Atomism in Late Medieval Philosophy and Theology

Atomism in Late Medieval

Philosophy and Theology

History of Science and Medicine Library

VOLUME 8

Medieval andEarly Modern Science

Editors

J.M.M.H. Thijssen, Radboud University NijmegenC.H. Lüthy, Radboud University Nijmegen

Editorial Consultants

Joël Biard, University of ToursSimo Knuuttila, University of HelsinkiJohn E. Murdoch, Harvard University

Jürgen Renn, Max-Planck-Institute for the History of ScienceTheo Verbeek, University of Utrecht

VOLUME 9

Atomism in Late Medieval Philosophy and Theology

Edited by

Christophe Grellard and Aurélien Robert

LEIDEN • BOSTON2009

ISSN 1872-0684ISBN 978 90 04 17217 3

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printed in the netherlands

On the cover: Goussin de Metz, l’Image du monde, Paris, 1304; Bibliothèque de Rennes Métropole, MS0593, f. 64a. Courtesy of the Bibliothèque de Rennes Métropole.

This book is printed on acid-free paper.

Library of Congress Cataloging-in-Publication Data

Atomism in late medieval philosophy and theology / edited by Christophe Grellard and Aurélien Robert. p. cm. — (History of science and medicine library ; v. 8) Includes bibliographical references and index. ISBN 978-90-04-17217-3 (hardback : alk. paper)1. Atomism. 2. Philosophy, Medieval. I. Grellard, Christophe. II. Robert, Aurélien. III. Title. IV. Series.

BD646.A85 2009 189—dc22

2008042420

CONTENTS

Preface ......................................................................................... viiList of Authors ............................................................................ ix

Introduction ................................................................................ 1 Christophe Grellard & Aurélien Robert

Beyond Aristotle: Indivisibles and Infi nite Divisibility in the Later Middle Ages .................................................................. 15 John E. Murdoch

Indivisibles and Infi nities: Rufus on Points ................................ 39 Rega Wood

Richard Kilvington on Continuity ............................................. 65 Elżbieta Jung & Robert Podkoński

The Importance of Atomism in the Philosophy of Gerard of Odo (O.F.M.) .......................................................................... 85 Sander W. de Boer

Nicholas of Autrecourt’s Atomistic Physics ............................... 107 Christophe Grellard

William Crathorn’s Mereotopological Atomism ........................ 127 Aurélien Robert

An Indivisibilist Argumentation at Paris around 1335: Michel of Montecalerio’s Question on Point and the Controversy with John Buridan .................................................................. 163 Jean Celeyrette

John Wyclif ’s Atomism ............................................................... 183 Emily Michael

Blasius of Parma facing Atomist Assumptions .......................... 221 Joël Biard

Bibliography ................................................................................ 235

Index of Ancient, Medieval and Renaissance Authors ............. 247Index of Modern and Contemporary Authors ......................... 249

vi contents

PREFACE

Most of the papers collected in this volume are the result of a confer-ence held at the Maison française d’Oxford in November 2004, which was organized by the present editors. This two-days workshop was entitled “Atomism and its Place in Medieval Philosophy” and its fi rst aim was to assess the different issues in which atomism could have played a role during the Middle Ages. But the contributions generally focused their target on the physical/mathematical distinction within medieval debates about the continuum and the indivisible. For this reason, each chapter has been thoroughly rewritten and we asked other scholars to participate to this book. This is the reason why we have decided to change the title for the publication with an even more general title.

As organizers of the conference, we would like to express our gratitude to the institutions that sponsored us and helped us with their fi nancial support: fi rst, the Maison française d’Oxford, where the workshop took place and whose director, Alexis Tadié, generously offered excellent conditions for the organization; the “Service Science et Technologie” of the French Embassy in Great Britain (London); the GDR 2522 “Philoso-phie de la connaissance et de la nature au Moyen Âge et à la Renais-sance” (CNRS, Tours); the ACI “Articulations entre mathématique et philosophie naturelle (XIV e–XVIe s.)” (CNRS) which all fi nanced the main part of the colloquium; fi nally, the Center “Tradition de la pensée classique” (EA 2482) of the University of Paris I Panthéon-Sorbonne who helped both for the conference and for the publication of the volume. We therefore thank them all for their participation, without which this conference and this book wouldn’t have existed.

Above all we must thank Alexis Tadié and Stéphane Van Damme who encouraged us and made the realization of this project possible. Our gratitude also goes to Margaret Cameron and Dallas Denery Junior who helped us to translate into English some of the chapters presented here which were initially written in French. We must also thank the people who were present at the conference but who didn’t take part in this volume: Richard Cross, Luc Foisneau, Gabriele Gal-luzzo, Dan Garber, Isabel Irribaren, Andrew Pyle, Sabine Rommevaux, Cecilia Trifogli.

LIST OF AUTHORS

Joël Biard, Université François Rabelais, Centre d’Études Supérieures de la Renaissance (UMR 6576 du CNRS), Tours, France.

Jean Celeyrette, UMR 8163 Savoirs, Textes, Langage CNRS-Université de Lille III, Lille, France.

Sander W. de Boer, Radboud Universiteit Nijmegen, Nijmegen, The Netherlands.

Christophe Grellard, Université de Paris I Panthéon-Sorbonne, Paris, France.

El bieta Jung, University of ŁódΩ, ŁódΩ, Poland.

Emily Michael, Brooklyn College and the Graduate Center, City University of New York, New York, USA.

John E. Murdoch, Harvard University, Cambridge, Mass., USA.

Robert Podko ski, University of ŁódΩ, ŁódΩ, Poland.

Aurélien Robert, Centre d’Études Supérieures de la Renaissance (UMR 6576 du CNRS), École française de Rome, Tours-Roma, France-Italy.

Rega Wood, Stanford University, USA.

INTRODUCTION

Christophe Grellard & Aurélien Robert

1. Medieval Atomism in Recent Historiography

In the second half of the twentieth century, there has been a great renewal in the history of medieval atomism. If the rising of philosophi-cal debates on the composition and the divisibility of a continuum in the Latin West during the thirteenth and the fourteenth centuries had long been considered a mere transition between Ancient and Renaissance atomism,1 more recent studies have taken the opposite direction and tended to restore the image of a period of intense refl ections on indi-visibles. John E. Murdoch is mostly responsible for this turn in recent historiography, thanks to his work on a lot of still unedited authors.2 In some respects, this attitude is not that new if we consider that at the end of the nineteenth century, Kurd Lasswitz’s essay Geschichte der Atomistik vom Mittelalter bis Newton (1890) and Léopold Mabilleau’s Histoire de la philosophie atomistique (1895) had already attempted to make room for medieval atomism, though both of them were unfamiliar with most of the important authors of the fourteenth century, with the exception of Nicholas of Autecourt.3 But it is only recently that some important monographs have endeavoured to replace these two pioneering books. Bernhardt Pabst, in 1994,4 and Andrew Pyle in 1995,5 furnished new and detailed studies of medieval atomism, with full chapters dedicated to fourteenth-century atomism.6 The ambition of this book is not to replace those essays in the history of medieval atomism, but to raise some new questions about the commonly accepted view of the nature

1 We should mention earlier monographs on this topic, such as Van Melsen, From Atomos to Atom, where the chapter on medieval atomism is joined to the Renaissance period and is limited to developments on minima naturalia.

2 See the bibliography at the end of the volume.3 In fact, they only knew Autrecourt’s articles condemned in Paris.4 Pabst, Atomtheorien des lateinischen Mittelalters.5 Pyle, Atomism and its Critics from Democritus to Newton.6 For a more detailed historiographical essay on medieval atomism, cf. Lüthy,

Murdoch & Newman, Late Medieval and Early Modern Corpuscular Matter Theories, “intro-duction,” pp. 1–17.

2 christophe grellard & aurélien robert

of fourteenth-century atomism in particular. The aim is to go fur-ther into detail about specifi c authors than do the expansive histories mentioned above, though we do not intend the present work to be a mere collection of single essays on single authors. As this introduction will try to show, several leitmotivs are present throughout the different chapters. The principal tasks of this book are, fi rst, to distinguish the singularity of fourteenth-century atomism, compared to other periods; second, to show that the understanding of the debates over this period is far more complicated than it is usually asserted in the old as in the recent historiography; and third, to ask whether fourteenth-century atomism is rather mathematical, physical or even metaphysical, as some of the contributors have tried to challenge the prevailing view about the mathematical nature of indivisibilism at that time.

It is a diffi cult task to catch the essence of medieval atomism—if it exists—for as Gaston Bachelard used to say, the atomist doctrines become more and more confused when one wants to embrace them as a whole.7 It is undeniably true that different forms of atomism existed in the Middle Ages, from the Arabic occasionalist theories of the Mutakallimun to the infancy of modern science in the natural philoso-phy of the Renaissance. One may be tempted to distinguish the theory of the elements in the medical context, the medieval interpretations of Plato’s atomism in the Timaeus, the reconstruction of Democritus’s thesis through Aristotle’s critics, etc. For medieval attempts to consider the existence of atoms or indivisibles are to be found in very differ-ent contexts. In any case, the rather common attitude nowadays is to assert that apart from Arabic theological atomism, the only survival of atomism in the Middle Ages is the fourteenth-century “indivisibilism.”8 This way of dividing the history of atomism is certainly artifi cial and misleading, for as early as the twelfth century some atomistic theories of matter were developed by philosophers such as William of Conches. So we are left with Bachelard’s opinion, while historians tend to restrict themselves to their own specifi c areas. According to John E. Murdoch, for example, fourteenth-century atomism presents some particular features that allow the historian of philosophy and science to isolate this period from other traditions. The mathematical—or rather geo-metrical—nature of the fourteenth-century debates on the continuum

7 Bachelard, Les intuitions atomistiques, p. 11.8 Murdoch, “Naissance et développement de l’atomisme au bas Moyen Âge latin.”

introduction 3

and indivisibles differentiates it from William of Conches’s atomism, for example, which is in turn infl uenced by Plato’s Timaeus and by the medical school of Salerno and has nothing to do with Aristotle’s geo-metrical attacks against Leucippus and Democritus.

In this volume, we will focus on fourteenth-century discussions of indivisibles and atoms,9 in order to take stock of the situation on recent historiography and above all to discuss Murdoch’s hypothesis, which is the prevailing one today. Indeed, all the chapters presented here try to respond, implicitly or explicitly, to the question: are debates on atomism in the fourteenth century purely mathematical and geometrical?

In order to give the reader a general idea of the history of atomism in the Middle Ages, let us fi rst describe in a few lines the Ancient sources that were available to the medieval philosophers, since it is partly at the origin of the main stream in recent historiography.

2. Ancient and Medieval Atomism

It has long been thought that Ancient atomists, such as Epicurus or Lucretius, were rediscovered during the Renaissance, notably after the works of Poggio Bracciolini on the manuscript of the De natura rerum discovered by him in 1417. But the manuscripts on which Poggio based his edition are from the 9th century, and we know now that Lucretius was still copied during the Middle Ages, as is evident from the many manuscripts of the De natura rerum dating after the 9th–10th centuries.10 Moreover, as J. Philippe has shown in his pioneering study,11 Lucretius’s poem was discussed throughout the Middle Ages with no interruption from the era of the Church Fathers to the twelfth century. An example of this persistence is William of Conches, who quotes a passage from the De natura rerum in his Dragmaticon philosophiae, though he didn’t have access to the original text but only knew it from Cicero, Priscian and probably other sources.12 The same is also true for Epicurus, whose works were known through a still longer chain of intermediate sources. One feels this tradition in the medieval encyclopaedias, as in Isidore of Seville, the Venerable Bede and Rhaban Maur, all of whom

9 One exception is Rega Wood’s chapter on Richard Rufus, which serves to clarify the particular situation of fourteenth-century discussions.

10 Philippe, “Lucrèce dans la théologie chrétienne,” (1896), pp. 151–152.11 Ibid.12 William of Conches, Dragmaticon philosophiae [ Ronca], p. 27.


discussed the existence of atoms, a tradition that continues up to Vincent of Beauvais, for example. John of Salisbury also dealt with Epicurism in his Metalogicon13 and in his Entheticus, where he tried to refute its principal tenets.14 We may multiply the examples, but there is no doubt that ancient atomist theories were known to the medieval philosophers and associated with names such as Epicurus or Lucretius. The idea of an eclipse of atomism seems to be confi rmed by the violent reactions of the Church Fathers. Everybody has in mind Lactantius’s attacks, for example.15 So, even if Augustine said that Epicurism was dead in his times, it is now well-known that this view has no basis in historical real-ity. It is true, indeed, that Ancient atomists were not discussed for their theories of matter as such, but rather for the theological consequences of their views, among them the negation of Divine providence and the implication of the impassivity of God. Therefore, despite the fact that many indirect sources were present during the Middle Ages, there were no new atomist theories of matter, nor detailed exegesis of ancient ideas, until the 12th century. One of the main reasons for this absence was the assimilation of Epicurism with heresy; and even if atomism is not necessarily connected with the theory of pleasure, its view of matter has been discredited for several ancillary reasons as well.16 Further, the main theses of Ancient atomism were known through severe critics or through partial quotations.17 The fi rst philosophical resurgence of atom-

13 Cf. John of Salisbury, Metalogicon [ Hall ], II, 2, 10–11, p. 58 or IV, 31, 22–27.14 The relevant passages are found in Philippe, “Lucrèce dans la théologie chré-

tienne,” (1896), pp. 158–159.15 For example: Lactantius, De ira dei, 10, 1–33.16 For example, in the 9th and 10th centuries, Epicurism was attached to the Cathar

heresy. Cf. Philippe, “Lucrèce dans la théologie chrétienne,” (1896), pp. 148–160. In the 11th century, one can still fi nd violent critics against Epicurus in Marbode of Rennes, Liber decem capitulorum [ Leotta], pp. 54–58. The passage ends with a terrible judgement (p. 58):

Quapropter stultos Epicuri respue sensus, Qui cupis ad vitam quandoque venire beatam; Sperne voluptates inimicas philosophiae, In grege porcorum nisi mavis pinguis haberi Illisa rigidam passurus fronte securim.17 As J. Philippe asserts (Ibid. p. 161): “Les citations de Lucrèce chez les gram-

mairiens, les extraits de son oeuvre donnés par les Apologistes assurèrent, bien mieux qu’un enseignement méthodique, la conservation de ses idées qui entrèrent ainsi dans l’enseignement théologique. Présenté comme système, l’Epicurisme eût été vite proscrit, et, de fait, il l’a été souvent: mais des citations éparses semblaient moins dangereuses, et, comme les idées qu’elles contenaient répondaient souvent à des questions soulevées par les commentaires bibliques, on les adopta sans défi ance.”

introduction 5

ism dates from the 12th century, in the works of William of Conches, but it is a result of different traditions, Platonist and medical, and it is not a strict reading of Ancient atomism.

In the 13th century, there were no developed atomist theories as far as we know—be they mathematical or physical—even if one can fi nd some corpuscular tendencies in some physicians as Urso of Salerno at the very end of the 12th century,18 or in some philosophers such as Robert Grosseteste and Roger Bacon.19 All of these authors discussed the concept of minimum, sometimes considering it as a synonym of “atom,” as is explicitly affi rmed by Albert the Great, for example.20 Not only there were no atomist theories of matter, but providence, indifference of God, hedonist visions of happiness, etc. were no longer subjects of discussion when authors treated the nature of matter, except for some detailed critiques of Arabic atomism in Thomas Aquinas’s Summa contra gentiles for example.21 The target had changed.

In addition to the theological motive for the eclipse of Ancient atomism, the arrival of new Latin translations of Aristotle’s texts on natural philosophy in the thirteenth century played an important role. Epicurus and Lucretius were no longer the protagonists when people dealt with the nature and function of atoms, as both were progressively replaced by Democritus, to whom Aristotle devoted many passages in the Physics, the De generatione et corruptione and De caelo. This is the reason why, according to Murdoch, fourteenth-century atomism is merely a response to Aristotle’s anti-atomism and never a return to Ancient theories. He wrote:

18 Cf. Jacquart, “Minima in Twelfth-Century Medical Texts from Salerno.”19 Cf. Molland, “Roger Bacon’s Corpuscular Tendencies (and some of Grosseteste’s

too).” Work should be done on David of Dinant, who seems to have been tempted by corpuscular theories too.

20 Albert the Great, De generatione et corruptione [ Hoddfeld ], t. 1, c. 12, p. 120, 44–55: “Democritus autem videbat quod omnia naturalia heterogenia componuntur ex simili-bus sicut manus ex carne et osse et huiusmodi, similia vero componuntur secundum essentiam ex minimis quae actionem formae habere possunt, licet enim non sit accipere minimum in partibus corporis, secundum quod est corpus, quod autem non accipi minus per divisionem, tamen est in corpore physico accipere ita parvam carnem qua si minor accipiatur, operationem carnis non perfi cet, et hoc est minimum corpus non in eo quod corpus, sed in eo quod physicum corpus, et hoc vocavit atomus Democritus.”

21 Cf. Aquinas, Summa contra Gentiles, III, c. 65 and 69. For the references to the Arabs in Aquinas, cf. Anawati, “Saint Thomas d’Aquin et les penseurs arabes: Les loquentes in lege Maurorum et leur philosophie de la nature.”


Unfortunately, almost all this indivisibilist literature is devoted to arguing against the Aristotelian position and to establishing that continua can be composed in this or that fashion of indivisibles; very little is said that helps to explain precisely why this current of indivisibilism arose in the fi rst third of the fourteenth century or what function it was held to serve. There seems to be no sign of a resurgence of ancient physical atomism among these late medieval indivisibilists, nor anything resembling a consciously atomistic interpretation of mathematics.22

Murdoch’s statement is partly true, because every philosopher who wrote something in the area of natural philosophy at this time had to discuss the question of continuity as found in the sixth book of Aristotle’s Physics, and this is not the indivisibilist’s privilege. Moreover, Murdoch is perfectly right in saying that one cannot fi nd real hints of ancient physical atomism in this fourteenth-century indivisibilist literature, for if there were physical theories for the existence of atoms, they had nothing in common with Democritus’s or Lucretius’s views on the sub-ject.23 However, can we limit our characterization of fourteenth-century atomism to its mathematical features?

3. Fourteenth-Century Atomism: Mathematical, Physical or Metaphysical?

Much recent research is devoted to showing that there was also a more physical form of atomism in the fourteenth century. In this respect, the most representative philosopher of this physicalist way of thought is undoubtedly Nicholas of Autrecourt, who is admittedly considered as an exception in the philosophical landscape of later medieval philosophy (see Christophe Grellard’s chapter). But there are other, lesser-known thinkers who developed consistent views about the physical nature of atoms and their role in the explanation of natural phenomena, among them Gerard of Odo, William Crathorn or John Wyclif, to whom some of the chapters in the present volume are dedicated (see the contribu-tions of De Boer, Robert and Michael).

22 J.E. Murdoch, “Infi nity and Continuity,” p. 576.23 It is clear that medieval philosophers knew mainly the physical part of Democritus’s

doctrine, because of the mediation of Aristotle’s critics (they also knew some epistemo-logical elements from Cicero and Aristotle’s Metaphysics, book Γ ). But it is also evident that Democritus’s thought cannot be limited to such an aspect. For recent presentations on different facets of his philosophy, cf. Brancacci & Morel, Democritus: Science, the Arts, and the Care of the Soul.

introduction 7

In this context of hesitation concerning the different possible streams of medieval atomism—physical or mathematical—the main aim of this book is to assess past and present research, focussing on the different forms taken by indivisibilist theories in the Latin West, as presented by their followers and their critics. It will appear that even at the end of the thirteenth century, when no indivisibilist theory was formed, the question of the nature of points was neither purely mathematical or geometrical, nor purely physical. Richard Rufus is a good representa-tive of such a mixed point of view (see Wood’s chapter). And, in the fourteenth century, both attitudes can be found. All the divisibilists used geometrical arguments, because they are much stronger than any other. On the contrary, some indivisibilists tried to show that geometry is not the right tool to argue against atomism if one considers atoms in a physi-cal or metaphysical way. Others tried to respond to the mathematical arguments, but always with physical considerations. With the exception of Michel of Montecalerio24 and Henry of Harclay, who discussed more precisely the mathematical arguments, Walter Chatton, William Crathorn, Gerard of Odo, Nicholas of Autecourt, and John Wyclif, demonstrate a strong propensity for the use of physical or metaphysical considerations. According to them, indivisibles must be considered as elemental components of reality, and not as mere unextended points. Nicholas of Autrecourt and William Crathorn even tried to develop a real atomistic physics, and John Wyclif ’s position could be traced back to the Platonist tradition inaugurated by the twelfth-century commenta-tors of Plato. In any case, their positions are never reducible to a mere reaction to Aristotle’s arguments, nor to a reconstruction of Democritus through Aristotelian doxography.

It is clear that from the divisibilist side, the strongest arguments against atomism are geometrical. They are presented by Aristotle, but also by Al-Ghazali’s and Duns Scotus’s works. As other examples of this attitude, we can mention Thomas Bradwardine, who dealt with the problem of the continuum in a mathematical and geometrical fashion, and Richard Kilvington, who tried to mix mathematical elements with physical thought experiments. Other divisibilists, as Walter Burley and Blasius of Parma, remained suspended between both methods.

Beyond the oppositions between indivisibilists and anti-indivisibilists, the aim of the present book is thus to give an account of the complexity

24 On Montecalerio, see Celeyrette’s chapter in this volume.


of the refl ections upon the structure of continuum and matter in medieval natural philosophy. Indeed, a strict opposition between two antagonistic sides would be exaggeratedly schematic, even if some personal struggles cannot be excluded.25 The chapters presented here may be considered a starting point for further studies about the dif-ferent atomist traditions in the Middle Ages, about the sources of medieval atomism, and about the relevant periods that must be taken into account by the historian of sciences.26 We hope that the follow-ing contributions will contribute to an understanding of atomism as a continuously—though more or less accurately—present context in medieval speculations about nature.

4. Overview of the Contributions

In his inaugural chapter, John E. Murdoch gives a general survey of the divisibilist/indivisibilist debates in the later Middle Ages and a detailed dramatis personae of atomists and their critics. It is argued that the real motive behind fourteenth-century indivisibilism remains the rejection of Aristotle’s arguments as formulated in his Physics and his other trea-tises on natural philosophy. But Murdoch also endeavours to show that medieval philosophers went far beyond Aristotle, though he is always the point of departure for their enquiries. Many of them put forth new elements and new methods for the analysis of the continuum’s divisibility that were by no means present in Aristotle’s texts. Murdoch insists that the major issues of this renewal were mathematical and geo-metrical, even if one can also fi nd a new language of analysis derived from logic in such a context.

This fi rst chapter can be considered as a guide through these com-plicated discussions, written from a standpoint representative of the prevailing historiography that some of the other chapters in this volume will challenge.

25 On this point see the dramatis personae in Murdoch’s contribution to this volume. 26 For example, studies about quantity in the twelfth century would probably

reveal some atomistic preoccupations, as is clear from Peter Abelard’s discussion in the Dialectica, where he detailed the theory of his master (William of Champeaux?), who clearly stated that quantities (lines, but even bodies) are made of indivisibles. Cf. Peter Abelard, Dialectica [ De Rijk], pp. 56–60. Therefore, we may fi nd some degree of similarity between twelfth and fourteenth-century indivisibilism. This would deserve another volume.

introduction 9

In Rega Wood’s chapter, one will fi nd a sketch, from a thirteenth-century standpoint, of what would become the main blind alleys for fourteenth-century philosophers. Are mathematical, physical and metaphysical points of view on the nature of points alike and are they directed to the same object? Rega Wood thus presents the interpreta-tion of Aristotle’s arguments against atomism by the thirteenth-century philosopher Richard Rufus of Cornwall, who dealt with this issue in many of his works. Since it is usually stated that medieval discussions about indivisibles are nothing but a mere reaction to the rediscovery of Aristotle’s texts, the case of Rufus seems important, for he belongs to the fi rst generation of philosophers who commented on the whole Aristotelian corpus. This chapter then shows how confused Aristotle’s positions were about the defi nition of point in his different books, and how this confusion could lead to different kinds of interpretations. The central question in Rufus’s works is to know whether points are mere mathematical objects or substances of some sort, since, surprisingly enough, both assertions can be found in Aristotle. Thus, Rega Wood thoroughly examines the texts in which Rufus distinguishes the respective roles and objects of mathematics, physics and metaphysics. Following Aristotle in the main lines, Rufus strongly denied that a continuum is composed of indivisibles from a mathematical point of view, but he admits that points are really found in sensible objects, from which mathematicians abstract their concept of point. Therefore, there is a strict link between mathematical and physical points. Moreover, in distinguishing sensible from intelligible matter, Rufus seems to contend that intelligible quantity is infi nitely divisible, while sensible matter has a kind of natural minimum. Turning then to the nature of points, Rufus considered them as accidents of matter, because matter can-not be spatially organized without the disposition of its points, i.e. by their respective positions. At the same time, he denies that points are constitutive parts of a body, even if they can be considered as a quasi cause of lines, for example. Therefore, points are primarily defi ned by their position in a line, a surface or a body, but are strictly extensionless. Hence, Rega Wood shows that points should be interpreted in Rufus’s thought as quasi essential parts of a line, but not as constitutive, nor as integral or quantitative parts of a substance. The fi nal section of the chapter deals with the question of the infi nite. If Rufus follows Aristotle in his criticism of physical atomism, all the same he affi rms that infi nities can be unequal, a position that sounds similar to the later view of Henry of Harclay.


In their contribution, Robert Podkoński and Elżbieta Jung present Kilvington’s curious attitude toward atomism. Indeed, even if he proposes some geometrical arguments for infi nite divisibility, he does not seem really interested in the classical debate as it appears in Duns Scotus’s reprisal of Avicenna’s rationes mathematicae. Neglecting the most popular arguments, he tries to elaborate new geometrical thought-experiments. Podkoński and Jung analyze carefully three of them: the fi rst deals with the angle of contingency, the second with the evolution of a triangle in a cone of shadow, and the last with the possibility of an infi nite line. In the two fi rst problems, Kilvington identifi es Euclid as the one who introduced the idea of an infi nitely small mathematical being, in the prop. 16 of the third book of the Elements, and Plato as his atomist opponent in the Timaeus, but he totally ignores all of his contemporaries. In all three cases, however, Kilvington is not really concerned with the confutation of atomism, even if he remains a fi rm defender of infi nite divisibility as an absolute principle. His fi rst aim is rather to examine and solve paradoxical cases linked to the question of continuum and to the Aristotelian thesis. On this point, he underlines two diffi culties against Aristotle: fi rst, it doesn’t seem possible to adopt the Archimedean and Euclidean principle of continuity; second, in an Aristotelian context, we are not able to answer Zeno’s paradox.

Despite these repeated attacks against indivisibilism, some authors tried to escape the threat of geometrical aporia by considering the problem from a more physical aspect. In his chapter, Sander W. de Boer endeavours to prove that Gerard of Odo (c. 1285–1349) is probably one of the fi rst consistent atomists in the fourteenth century. Of course, Henry of Harclay and Walter Chatton were indivisibilists before him, but they did not defend a physicalist point of view. From several unedited texts, Sander W. de Boer shows that atomism occupied a much more important place than past commentators usually assumed, and that Odo made new ontological claims about the indivisibles, to the effect that they are kinds of physical parts of the continuum. Odo’s atomism seems to rest on two basic claims: 1) indivisibles are parts prior to the whole they belong to; and 2) it follows from the mereological assertion in 1) that the number of atoms should be fi nite. To establish these two claims, Odo calls in different physical phenomena, such as the nature of the degrees of heat, intension and remission of light, etc. Rejecting the existence of actual infi nities, Odo contends that there must be minima and maxima in natural phenomena. Sander W. de Boer concludes

introduction 11

that Odo “consistently uses his atomism in explaining reality, and the application of this atomism to God’s power and to the inner structure of continuous physical processes is not provoked by any mathematical arguments.”

More famous than Gerard of Odo is Nicholas of Autrecourt, who is undoubtedly one of the most studied of the fourteenth-century atomists. In his chapter, Christophe Grellard demonstrates that Nicholas did not limit himself to considerations about the existence and nature of indivisibles, but rather that he explored the possibility of forming a complete alternative physics. The main concern of Nicholas was to prove the eternity of the world from atomistic explanations of generation and corruption, i.e. aggregation and segregation, of atoms which eternally exist. Of course, Autrecourt’s standpoint is physical in this context when he constructs the conditions of a local motion in a void, or when he explains condensation and rarefaction and other natural phenomena. But his attitude is also partly metaphysical, when he criticizes, for example, the matter/form couple in order to reduce matter to a mere atomic fl ux. As Christophe Grellard shows, there is no need of a substratum in change according to Autrecourt; rather, the atomic fl ux is enough because atoms are not bare particulars, but qualitative entities. They are the basic substances of the world. In some respect, these atoms are more similar to Aristotle’s minima naturalia than to Democritean atoms. How then to explain the unity of a thing composed of atoms? Nicholas takes for granted that there are essential and accidental atoms in a natural compound. The essential ones func-tion as kinds of natural magnets and make the others hold together. Indeed, even if he criticizes Aristotle’s distinction between matter and form, such essential atoms are sometimes called ‘formal atoms’, in the sense that they contain what will be the principle of motion, a sort of virtus. But, since the natural power of these atoms is not always suffi cient, an atomic compound can be helped by a celestial infl uence. Surprisingly enough, this copulatio between atoms and stars is a sine qua non condition for a natural change. For example, procreation requires three conditions: it occurs through a material condition (the sperm), a formal condition (the man) and by the effi ciency of a star. Autecourt’s cosmology is therefore purely atomistic, from the explanation of the celestial stars to the natural phenomena in the sensible world. This genuine theory, as Christophe Grellard shows, takes its origin not only in some developments of Aristotle’s corpus itself, especially from the


De generatione et corruptione and the De caelo, but also from the Arabic atomism of the Mutakallimun, known through the Latin translation of Maimonides’s Guide of the perplexed.

At the very same time, around 1330, another philosopher, William Crathorn, developed a similar theory, minus the celestial action on atomic compounds. Aurélien Robert’s aim is to show that Crathorn puts forth the basic foundations for an atomistic physics, which rests upon two distinctive features: a mereological interpretation of the continuum debate, and a systematic use of the notions of space and position. In Crathorn’s theory, indivisibles are conceived as things, actually exist-ing in the continuum as real parts, and occupying a single place in the universe. Moreover, these entities have a certain nature (there are atoms of gold and atoms of lead, for example) and it must be inferred, as Robert shows, that they also have a certain quantity or magnitude. From this reconstruction of the physical or metaphysical structure of atoms, it is demonstrated that Crathorn applied this theory to Aristotle’s arguments, giving a new defi nition of contiguity and continuity from the arrangement of parts and from the contiguity of places occupied by the atoms. Hence, Crathorn reduces all movement to a local motion of atoms, and explains various physical phenomena with his new analysis of the indivisibles (such as condensation and rarefaction, for example). Finally, even if Crathorn’s attempt to elaborate an atomistic physics is rather original, Aurélien Robert brings out some limits to his analysis due to theological reasons, notably when the Oxford master applies his analysis to the cases of angels or souls. In conclusion, this tends to prove that from mereotopological elements, Crathorn endeavoured to think the possibility of an atomistic physics, where atoms resemble more the minima naturalia than the atoms of Democritus, Epicurus or Lucretius.

Jean Celeyrette takes up a lesser-known Parisian dispute between John Buridan and Michel of Montecalerio which took place a short time before and after 1335. Buridan’s question De puncto has been edited by Vladimir Zubov in 1961, but Jean Celeyrette makes its context far more clear for us. Indeed, he provides a new evidentiary basis for reconstructing the whole debate from unedited manuscript material. It appears that the departure point of the confrontation between both masters was a discussion of the Ockhamist view about points. Though Buridan doesn’t agree with Ockham in his commentaries on Aristotle’s Physics, he followed him in his De puncto. Jean Celeyrette then details the different steps of the dispute, showing that there were probably four consecutive disputed questions—and possibly four texts—in which both

introduction 13

masters responded to each other. Montecalerio presents an indivisibilist theory very different from the ones examined in the other chapters of this volume, for he doesn’t want to identify points with parts, as Odo, Autrecourt and Crathorn tend to do. Points are considered as accidents existing in a substance as in a subject (subiective), a position similar to Rufus’s view. Incidentally, this chapter presents an interesting standpoint for the general historiographical question posed in this book about the mathematical or physical nature of atomist debates in the Middle Ages, for Jean Celeyrette concludes: “. . . mathematics are roughly absent . . . One fi nds no allusion, even to challenge their relevance, to the rationes mathematicae very fashionable among English scholars since Scot.” This chapter even demonstrates an appeal to physical practice in Buridan’s text—though he is a divisibilist—when he invokes, for example, experi-mentation and the work of the Alchemists. Therefore, Celeyrette’s study tends to establish the fact that even in a discussion about the possible existence of points, some medieval thinkers used to consider it not as a purely mathematical problem, but also as a real need for physics.

The last atomist philosopher presented in this volume is John Wyclif. The main purpose of Emily Michael’s chapter is to examine how Wyclif tried to make an atomistic view of prime matter compatible with a hylomorphist conception of natural beings. As Michael shows, in Wyclif ’s cosmology, matter is fi rstly conceived as a composition of a fi nite number of atoms defi ned as real entities occupying each pos-sible place in the world. Though indivisible parts of matter have no quantity, nor quality, their contiguity in space defi nes the total shape and quantity of matter in the world. This theory is very similar to Crathorn’s position, except that there is no void in the world accord-ing to Wyclif. If one turns to the reasons for Wyclif ’s adoption of such a corpuscular theory of matter, Michael contends, one should fi nd that his fi rst motivation was theological, for his view is supported by a logical interpretation of Scripture. Wyclif asserts that God has created a fi nite world with a fi nite number of atoms in it. Michael thus demonstrates that the whole methodology of Wyclif ’s cosmology is directed by the logic of Scripture and by the interpretation of the Book of Genesis. Afterwards, Michael turns to the compatibility of the atomistic view of matter with a pluralistic hylomorphism, inspired by some of his scholastic predecessors, especially in the Franciscan school. Finally, Wyclif ’s atomism is evaluated from the question of the mixtio of the elements. The traditional view of Aquinas and other scholastics was that in the generation of a new compound, its elements do not remain in the mixture. On the contrary, Wyclif thinks that the elements remain


in the compound as small atomic particles. As a result of her analysis, Michael establishes that what Wyclif calls minima naturalia comes from such a natural mixing process of the elemental atoms. Therefore, there is a natural hierarchy of beings: indivisibles, i.e. extensionless points; the elemental atoms (air, earth, fi re and water), which are determined by an elemental form; minima naturalia (minimal parts of fl esh, bones, etc.), which are the fundamental particles of a body composed of elemental atoms; bodies, formed by the composition of those minima naturalia plus substantial forms; and human beings, which are bodies plus the rational soul. This chapter brings us a new, important view of a non-mathematical but metaphysical form of indivisibilism which takes its place in a whole cosmology inspired by Plato’s Timaeus and Augustine’s De Genesi ad litteram.

With Joël Biard’s contribution on Blasius of Parma’s attitude towards atomism, we reach the very end of the fourteenth century. It is worth noting that Blasius reveals the permanency of atomistic problematic, even when most atomist philosophers seem to have disappeared. Indeed, from time to time, the Italian philosopher seems ready to use a kind of atomistic point of view. Leaving aside some aspects of Blasius’s thought (the question of the void and of the nature of matter) Biard deals pri-marily with the two main questions on the continuum and on the minima naturalia. About the fi rst point, Blasius fi rst denies the possible existence of indivisibles, by using classical geometrical arguments. Nevertheless, he seems to admit as rather plausible a kind of indivisibilism, that is infi nite indivisibilism. This partial defense of atomism relies on both mathematical and physical arguments. In sum, Blasius tries to accept simultaneously the infi nite divisibility of a line and the infi nity of points. Without quoting any atomists (such as Henry of Harclay or Nicholas of Autrecourt, both of whom accept the existence of an infi nity of indivisibles), Blasius fi nally defends a kind of indivisibilism, but it is going too far to consider him as an atomist. Indeed, concerning the question of natural minima, he clearly asserts the infi nite divisibility of matter. But, once again, atomism implicitly remains: if there is no absolute minimum, Blasius concedes, there should be a minimum in the sense of physical limits of existence; and this limit is determined by the proportion of matter in a being. Finally, we should say that at the end of the century, atomist solutions were still more or less pres-ent as a convenient answer in some context between mathematics and physics. Blasius of Parma is an interesting witness to this kind of “regional atomism.”

BEYOND ARISTOTLE: INDIVISIBLES AND INFINITE DIVISIBILITY IN THE LATER MIDDLE AGES

John E. Murdoch

The basic text for late medieval Latin atomism and its critics was Aristotle’s Physics, especially Book VI. Here the atoms or indivisibles he considered and combatted were extensionless, a conception that can be found in scholastic debate about atoms all the way to Galileo and his atomi non quanti.1

The medieval atomists were clustered in the fourteenth-century,2 as were their Aristotelian critics. Figure 1 provides the basic dramatis personae of the fourteenth-century atomists and their critics. The list of atomists is nearly complete, save for the followers of Wyclif. The list of their critics is naturally less complete, being made up of chiefl y those who name their atomist opponents. Yet even without such identifi cation, we can often tell other critics, such as John Buridan and his school, because they oppose specifi c identifi able atomist arguments.

The question of the motives for the late medieval atomism is pretty murky. The motives for Greek atomism are, at least to some extent, an answer to Parmenides’s monism and center in attempts to explain natural phenomena (if not always totally successfully). Equally clear are the motives for the Arabic atomism of the Mutakallimun: namely, to put all causal relations into the hands of God through the mechanism of the doctrine of continuous creation. However, in the case of late medieval atomism there is not such a wholesale application to nature or to a God who creates the universe anew at every instant.

1 Galileo Galilei, Discorsi e dimostrazione matematiche intorno a due nuove scienze, vol. 8, p. 72. There is, of course, the quite separate consideration of minima naturalia that arises out of Aristotle’s criticism of Anaxagoras in Physics, I, ch. 4. For the medieval history, as well as the historiography, of this kind of atomism or corpuscularianism, see Murdoch, “The Medieval and Renaissance Tradition of Minima Naturalia”.

2 For the earlier medieval atomism by the likes of Isidore of Seville, William of Conches, etc., see Pabst, Atomentheorien des lateinischen Mittelalters. For the standard treat-ments of the medieval atomism of the fourteenth century, see Duhem, Le système du monde, vol. 7, pp. 3–157; Maier, “Kontinuum, Minima und aktuell Unendliches,” In Die Vorläufer galileis im 14. Jahrhundert 2nd ed., pp. 155–215; and, more briefl y, Murdoch, “Infi nity and Continuity.”

16 john e. murdoch

Dramatis personae

INDIVISIBILISTS ARISTOTELIANS

Henry of Harclay

Walter Chatton OFM

Crathorn OP

John Wyclif

ENGLISH

William of Alnwick OFM

Adam Wodeham OFM

Thomas Bradwardine

William of Ockham OFM

Roger Rosetus OFM

Walter Burley

John the CanonGerard of Odo OFM

Nicholas Bonetus OFM

John Gedo

Marcus Trevisano

Nicholas Autrecourt

CONTINENTAL

Single line arrows represent (named) criticismDouble line arrows represent verbatim borrowing

Fig. 1. Fourteenth-Century Indivisibilism and its Critics

indivisibles and infinite divisibility 17

The most frequently occurring “motive” is that of angelic motion, although it sometimes functions as an excuse to discuss at length the atomist or indivisibilist composition of continua.3 Alternatively, Henry of Harclay was convinced that the belief in the possible inequality of infi nites was grounds for a related belief in the composition of conti-nua out of indivisibles.4 Yet one feels that the real motive behind such fourteenth-century atomists was the scrutiny and consequent rejection of Aristotle’s arguments against such atomism or indivisibilism.

I now want to turn to my major topic: to measure how both the late medieval atomists and their critics went beyond (and not just developed) the Aristotelian base from which they began; beyond in the sense of providing new conceptions and new arguments for their cause.

Before that, however, I want to establish that, in coming up with the extensionless indivisibles of Book VI of Aristotle’s Physics, the fourteenth-century medieval atomists skewed the view of ancient atomism. This meant, of course, the opinion of Democritus, since they either did not have or failed to appeal to Epicurus or Lucretius. For example, citing Aristotle on Book I of De generatione, the late medieval indivisibilists

3 Walter Chatton, Reportatio Super Sententias [ Etzkorn e.a.] Liber II, pp. 114–146: “Et quia non potest sciri de motu angeli utrum sit continuus vel discretus in motu nisi sciatur utrum motus et alia continua componantur ex indivisibilibus, ideo quaero propter motum angeli utrum quantum componatur ex indivisibilibus sive permanens sive successivum.” Then Chatton spends the remaining 32 pages investigating this latter question and never returns to the notion of angelic motion. Similarly Gerard of Odo, Super primum Sententiarum, dist. 37 (MSS Naples, Bib. Naz. VII. B.25, ff. 234v–244v; Valencia, Cated. 139, ff. 120v–125v): “Ad quorum evidentiam querenda sunt quatuor . . . Tertium utrum motus angeli habeat partem aliquam simpliciter pri-mam.” But then on the very next folio he breaks into what is of real interest to him: “Utrum continuum componatur ex indivisibilibus;” he continues this inquiry to the end of the question, only devoting a brief paragraph at its end to the problem de motu angeli. Indeed, two other MSS of Gerard’s question, shorn of its concern about angelic motion, made it appear as if Gerard had written a work on Aristotle’s Physics. Both Walter and Gerard were atomists, but it is worth noting that Duns Scotus espoused an Aristotelian opposition to atomism (and, of course, chronologically preceding these two atomists) and included his discussion of the continuum in the context of angelic motion (Comm. Sent, II, dist 2, Q. 9), perhaps encouraging later atomists to do so (one should note that both Chatton and Odo were also Franciscans).

4 Henry of Harclay, “Utrum mundus poterit durare in eternum a parte post” (which amounts to his Quaestio de infi nito et continuo), MSS Tortosa, Cated. 88, ff. 87r–v; Florence, Bib. Naz. Fondo princ. II.II.281, fol. 97r: “Preterea, specialiter contra hoc quod dicitur in auctoritate Lincolniensis: Quod plura sunt puncta in uno magno continuo quam in uno parvo. Contra hoc sunt omnia argumenta que probant continuum non posse componi ex indivisibilibus; probant enim eciam quod in uno continuo non sint plura puncta quam in alio.”

18 john e. murdoch

maintained the essential agreement of Plato and Democritus with respect to the composition of continua out of indivisibles.5

Now the medievalists did not have Theophrastus on Democritus’s account of the variation of tastes through a corresponding variation in the shapes of atoms,6 but they did have the basic passage in the Metaphysics saying that atoms varied in shape, arrangement and posi-tion (shape being far and away the most important).7 And they also had the fourth chapter of De sensu where they might have guessed the Democritean account of tastes.8

Accordingly, the Aristotelian critics of medieval atomism correctly recognized that the indivisibles of Democritus have magnitude and have parts.9 Thomas Bradwardine, attempting to refute all sorts of atomism in his Tractatus de continuo, says rightly that Democritus held continua to be composed of indivisible bodies, though he devotes little space to this view and claims that what Democritus really had in mind was compo-sition out of an infi nite number of substances. Moreover, he does not properly refute the Democritean opinion since he sometimes takes it

5 Gerard of Odo, Super primum Sententiarum, I, dist. 37 (MSS citt. Note 3, Naples, fol. 235r; Valencia, fol. 120v): “Quantum ad primum sciendum quod, ut recitatur primo De generatione, opinio fuit Platonis et Democriti quod continuum componitur ex indi-visibilibus et resolvitur in indivisibilia. Diversimode tamen, quia Democritus asserebat corpus componi ex atthomis et resolvi in atthomas sive in magnitudines indivisibiles, quod idem est; Plato vero ponebat corpus componi ex superfi ciebus, superfi cies ex lineis, lineas ex punctis, et eodem modo resolvi. Convenienebant tamen in hoe quod uterque dicebat primam compositionem continui esse ex indivisibilibus et ultimam divisionem terminari ad indivisibilia.” Much later we fi nd a similar view expressed by the Wyclifi te John Tarteys in his Logica (MS Salamanca 2358, fol. 97v): “In ista materia, sicut in omni alia materia naturali et philosophica, est specialiter credendum illi parti pro qua ratio plus laborat inducendo nuda dicta Aristotelis sonantia in oppositum cum tunicis quas texerunt sapientes sequaces Platonis et Democriti qui convincerunt ex ratione infallibili corpora continua ex atthomis, id est, partibus indivisibilibus, integrari.” It is worth noting that Nicholas Bonetus claims to be following Democritus and opts for atomic magnitudes in each species of bodies, surfaces, lines, and points, although he does not cite the passage of Aristotle from De generatione I. On the whole medieval history of this passage (which some historians have held to be as much Aristotle as Democritus), see Murdoch, “Aristotle on Democritus’s Argument Against Infi nite Divisbility in De generatione et corruptione, Book I, Chapter 2.”

6 Theophrastus, De sensu, 49–83 (espec. 65).7 Metaphysics, A, ch. 4, 985b4–22.8 Aristotle, De sensu, ch. 4.9 For example, Albert of Saxony, Quaestiones in octo libros physicorum [Paris, 1518],

VI, q. 1, f. 64v: “Alio modo capitur indivisibile quod, licet habeat partem vel partes, tamen propter eius parvitatem non sunt abinvicem separabiles; talia indivisibilia posuit Democritus que vocavit corpora atomalia.” Nicole Oresme made a similar distinction among indivisibles. (Cf. his Quaestiones Physicorum, MS Sevilla 7–6–30, fol. 66r).


to be included in his extensive refutation of extensionless indivisibles composing a continuum, be they infi nite or fi nite in number.10

Let us return, however, to the examination of the new notions and arguments beyond the Aristotelian base of Physics VI.

One of the most obvious developments in this regard was the geometrical arguments added to support Aristotle’s opposition to indivisibilism. One source for them was clearly the Latin translation of Al-Ghazali’s Metaphysica.11 Fundamentally, these arguments against indivisiblism were based on techniques of parallel or radial projection which entail, if geometrical fi gures were composed of indivisibles, the equality, for example, of the sides of a square with its diagonal or of two concentric circles. Thus, if parallels were drawn between every indivisible or point in the sides of the square they would cut its diagonal in the same number of indivisibles or points, from which their equality followed. And the same, mutatis mutandis, for concentric circles where all radii are drawn. (Figure 2). These kinds of arguments gained in popularity and prestige when Duns Scotus used them in his own opposition to indivisibilism, even bringing Euclid into the picture as Al-Ghazali had not.12

The fourteenth-century atomists’ replies to these geometrical argu-ments were almost always highly unsatisfactory. Their answers often employed inappropriately physical ideas into geometry. Thus, Henry of Harclay, the fi rst clearly established atomist on the English scene, claimed that, like two sticks, the parallels between the sides of a square “take more” of the diagonal than they do of the sides.13 Or Walter

10 Thomas Bradwardine, Tractatus de continuo, MS Toruń, Poland, R402, p. 187 (the MS is paginated): “Omnes igitur opiniones erronee specialiter reprobantur, preter opinionem Democriti ponentem continuum componi ex corporibus indivisibilibus, que tamen per illam conclusionem et eius corollarium suffi cienter reprobatur. Non tamen est verisimile quod tantus philosophus posuit aliquod corpus indivisibile, sicud corpus in principio est diffi nitum, sed forte per corpora indivisibilia intellexit partes substantie indivisibiles et voluit dicere substantiam componi ex substantiis indivisibilibus.”

11 Cf. Al-Ghazali, Metaphysica [ Muckle], pp. 10–13. This is a Latin translation of Al-Ghazali’s Maqāsid al-falāsifa which in turn is taken from Avicenna’s Persian work Dānish-Nāmeh. For the most satisfactory translation of this latter, see Avicenne, Le livre de science [tr. Achena & Massé]. The medieval Latin scholars did not know these details and, in any case, did not have a Latin translation of this work of Avicenna’s.

12 John Duns Scotus, Comm. Sent. II, dist 2, q. 9. English translations are available of relevant passages in both Al-Ghazali and Duns Scotus in Grant, A Source Book in Medieval Science, pp. 314–319.

13 Henry of Harclay (MSS citt. note 4: Tortosa, 90v; Florence, 99r): “Voco autem ‘recte intercipi’ partem linee intercipi inter duas lineas equidistantes facientem cum lineis equidistantibus angulos rectos. Talis porcio est equalis in omni parte equidistancium

20 john e. murdoch

Chatton maintained that drawing all the radii of two concentric circles can not be done due to defectus materiae.14 Moreover, to account for the incommensurability of the diagonal and the side of a square, Harclay amazingly claims (even citing Euclid in the course of his argument) that the ratio of points in the diagonal and side is as two mutually prime numbers.15

One may contrast this with Bradwardine, who cites both Harclay and Chatton in his treatment of the continuum that systematically denies any indivisibilism anywhere, but highlights its inconsistency with mathematics, geometry in particular.16 Alternatively, John Buridan, in a treatment of indivisibilism not nearly so geometrical as Bradwardine’s, nevertheless claims that mathematical atomism would totally annihilate geometry.17 Bradwardine, however, goes a step further in asking whether in using Euclid’s geometry to upend indivisibilism he might be guilty of a petitio. In his questioning, he was in effect inquiring into, we would now say, the independence of axioms or suppositions.18

linearum. Nam illo modo solo debet intelligi equedistancia linearum. Si porcio linee oblique caderet inter lineas equidistantes, multo maior intercipitur quam alia recte cadens. Ita dico quod est de puncto. Nam inter lineas equidistantes et immediate se habentes non posset intercipi punctus secundum situm rectum, et tamen posset secun-dum obliquum. Et licet istud videtur mirabile de puncto, cum sit indivisibilis, tamen istud est necessario verum.”

14 Walter Chatton, Reportatio super Sententias [ Etzkorn e.a.], II, dist. 2, q. 3, p. 131: “Ad aliud de circulis dico unum generale, quod ubicumque oporteret dividere punc-tum, ponam lineam interrumpi et non procedere. Et quod arguitur contra eum per communem conceptionem positam, I Euclidis vel petitionem a puncto ad punctum quodcumque lineam ducere, negat illud propter defectum materiae in casu. Unde si protrahas unam lineam a puncto maioris circuli ad centrum et post velis protrahere aliam a puncto proximo mediato prius accepto, illa forte transibit vel veniet ad idem punctum minoris circuli; et ideo si velis protrahere tertiam a puncto intermedio duobus punctis prius acceptis, illa interrumpetur propter defectum materiae quando veniet ad concursum priorum linearum.”

15 Henry of Harclay (MSS citt., note 4; Tortosa, 91r; Florence, 99v): “Ad hoc potest dici quod Euclides in ista proposicione per quam probat quod diameter et costa sunt incommensurabilia, intelligit quod non numerantur communiter per aliquam unam quantitatem vel per aliquod per se divisibile; non quin numerentur per punctum. Et tunc est dicendum quod diameter et costa se habent sicut duo numeri contra se primi, qui non numerantur per aliquem numerum communem, set per solam unitatem.”

16 For a summation of Bradwardine’s procedure, see Murdoch, “Thomas Brad-wardine: mathematics and continuity in the fourteenth century.” This article gives the Latin text of the enunciations of the defi nitiones, suppositiones, and conclusiones of this work of Bradwardine’s.

17 John Buridan, Quaestiones in octo libros Physicorum [ Paris, 1509], VI, q. 2: “. . . et omnino perirent conclusiones et suppositiones geometrie.”

18 Thomas Bradwardine, Tractatus de continuo, (MS Toruń R.40.2, pp. 156, 188). Bradwardine fi rst mentions the possibility of a petitio in a comment relative to his fourth


Although Aristotle himself on occasion referred to mathematics and geometry in his analysis of continuous magnitudes, this was not central to his discussion. Instead, in Physics VI we fi nd him giving two conceptions or defi nitions of what it is for something to be continuous. First, things are continuous the parts of which have their extremities as one (ultima sunt unum).19 Secondly, everything that is continuous is divisible into divisibles that are always further divisible, that is, it is infi nitely divisible.20

To the latter notion fi rst: Aristotle in Book III of the Physics had given an extensive investigation of the infi nite and had distinguished between what came to be called an actual infi nite and a potential infi nite, hold-ing that the latter was the only kind that was permissible (a distinction that he does not bring up in his Book VI inquiry to the continuum). Yet there is a consideration not brought up at all by Aristotle, but whose investigation fairly bristles in the Middle Ages—especially in the

supposition (which reads: Omnes scientias veras esse ubi non supponitur continuum ex indivisibilibus componi): “Hoc dicit quia aliquando utitur declaratis in aliis scientiis quasi manifestis, quia nimis longum esset hec omnia declarare. Ubi autem tractant de compositione continui ex indivisibilibus non supponit eas veras esse propter petitionem principii evitandam.” He fi lls this out much later by maintaining: “Posset autem circa predicta fi eri una falsigraphia: Avroys in commento suo super Physicorum, ubi dicit, quod naturalis demonstrat continuum esse divisibile in infi nitum et geometer hoc non probat, sed supponit tanquam demonstratum in scientia naturali, potest igitur inpug-nare demonstrationes geometricas prius factas dicendo: Geometriam ubique supponere continuum ex indivisibilibus non componi et illud demonstrari non posse. Sed illud non valet, quia suppositum falsum. Non enim ponitur inter demonstrationes geometricas continuum non componi ex indivisibilibus nec dyalecticer indigent ubique, quoniam <non> in 5to Elementorum Eudlidis. Et similiter, nec geometer in aliqua demonstratione supponit continuum non componi ex infi nitis indivisibilibus mediatis, quia, dato eius opposito, quelibet demonstratio non minus procedit, ut patet inductive scienti conclu-siones geometricas demonstrare. Verumtamen Euclides in geometria sua supponit, quod continuum ex [in] fi nitis et immediatis athomis non componitur, licet hoc inter suas suppositiones expresse non ponat. Si falsigraphus dicat contrarium et ponat aliquam lineam ex duobus punctis componi, Euclides non potest suam conclusionem primam demonstrare, quia super huius lineam non posset triangulus equilaterus collocari, quia nullum angulum haberet, ut patet per 16am et eius commentum. Similiter, si dicat falsig-raphus, continuum ex athomis immediatis componi, 4am suam conslusionem et 8am non probat, ambe enim per su<per>positionem probantur. Similiter in probatione <23> 3i. Iste autem conclusiones non demonstrantur per aliquas conclusiones priores, sed ex immediatis principiis ostenduntur. Per has autem conclusiones relique demonstrantur, et ex his 3bus quasi tota geometria Euclidis dependet et in ipsa omnis alia geometria fundatur, quare geometria supponit <continuum> ex [in]fi nitis et immediatis athomis non componi.” For more on this problem of a petitio, see the article cited in note 16, pp. 117–119.

19 Aristotle, Physics, VI, 1, 231a22–29 (cf. Physics V, 3, 227a10–12).20 Aristotle, Physics, VI, 1, 231b12–15.

22 john e. murdoch

fourteenth century. This is whether or not there can be unequal infi -nites or, put another way, whether there can be infi nites that have a part/whole relation to one another. To put it yet another way, must all infinites be equal to one another?21 Initially the possibility (or impossibility) of unequal infi nites appeared relative to the problem of an eternal world. For instance, given a past eternity, there would be twelve times as many months as years, therefore, one infi nite would be twelve times another infi nite (and, consequently, there could not be an eternal world).22

Henry of Harclay, who allowed unequal infi nites and made it a central part of his investigation of the infi nite and the continuum, deserves credit for being apparently the fi rst to say that past eternity is the mirror of future eternity (which was theologically all right), and that any argument that could be made against past eternity would be effective against future eternity.23 And he also saw (as did many oth-ers) that unequal infi nites were involved in the infi nite divisibility of a continuum (for example, there were more parts in a whole continuum than in its half ).

21 There were other ancient sources for the notion of unequal infi nites, but they did not receive, one way or another, translation into Latin. See, for example, Plutarch, De comm. not. Adv. Stoicos, 1097a; Philoponus, De aeternitate mundi contra Proclum, I, 3, [Rabe], p. 11, and Apud Simplicium, Phys, VIII, 1, [ Diels], 1179, pp. 15–27; Alexander Aphrodisias, Quaest. natural., III, p. 12; Proclus, Comm. in Euclidem, Def. 17, Elem. theol., prop. 1; Lucretius, De natura rerum, I, 615–626. Some medievals imagined Aristotle to have something like the notion of infi nites having part/whole relations Phys. III, 5, 204a22–27 (but here clearly has in mind quidditative, not quantitative parts). Cf. for example, Walter Burley, Super Aristotelis libros de physica auscultatione commentaria [ Venice, 1589], coll. 288–289.

22 Archetypical is Bonaventure (Sent. II, dist. 1, p. 1, art. 1, q. 2): “Prima est haec. Impossibile est infi nito addi—haec est manifesta per se, quia omne illud quod recipit additionem, fi t maius; infi nito autem nihil maius, sed si mundus est sine principio, duravit in infi nitum: ergo durationi eius non potest addi. Sed constat, hoc esse fal-sum, quia revolutio additur revolutioni omni die: ergo etc. Si dicas, quod infi nitum est quantum ad praeterita, tamen quantum ad praesens, quod nunc est, est fi nitum actu, et ideo ex ea parte, qua fi nitum est actu, est reperire maius; contra, ostenditur, quod in praeterito est reperire maius: haec est veritas infallibilis, quod, si mundus est aeternus, revolutiones solis in orbe suo sunt infi nitae; rursus, pro una revolutione solis necesse est fuisse duodecim ipsius lunae: ergo plus revoluta est luna quam sol; et sol infi nities: ergo infi nitorum ex ea parte, qua infi nita sunt, est reperire excessum. Hoc autem est impossibile: ergo etc.”

23 Henry of Harclay (MSS citt. note 4: Tortosa, 82v; Florence, f. 94v): “Preterea, ad principale videtur quod eadem argumenta que probant mundum non potuisse fuisse ab eterno, eadem possunt fi eri ad probandum mundum non posse esse in eternum.”


Relative to the question of unequal infi nites, basically three tradi-tions were operative in the Middle Ages: (1) Unequal infi nites are not allowable and, therefore, any situation which implied them is equally untenable (most often such situation as the possibility of an eternal world).24 (2) Given that no infi nite is greater than another and yet that some infi nites are unequal, one concludes that infi nites are incompa-rable to one another.25 (3) Revise the rules about parts and wholes to include infi nite magnitudes and multitudes. This third alternative for the most part begins with Harclay (who fi nds a predecessor in Robert Grosseteste) and is carried further by William of Ockham and reaches its peak with Gregory of Rimini.26 From the standpoint of understanding

24 For example, Bonaventure (op. cit., note 22), Thomas Bradwardine, De causa Dei contra Pelagium, cap. 1 coroll. 40, espec. pp. 124–125. For many others, see Dales, Medieval Discussions of the Eternity of the World and Dales & Argerami, Medieval Latin Texts on the Eternity of the World.

25 This appears to be a fourteenth-century Parisian tradition. Nicole Oresme, Quaestiones super libros Physicorum, III, q. 12 (MS Sevilla Colomb. 7–6–30, ff. 37v–39v): “Utrum infi nitum sit alio maius aut equale sive minus vel utrum esset, si esset infi nitum, vel utrum infi nitum sit infi nito comparabile.” Albert of Saxony, Quaestiones in libros de caelo et mundo, I, q. 8 [Paris, 1518]: “Utrum infi nitum posit esse maius vel minus alio, si essent plura infi nita, seu utrum sit unum comparabile alteri.” This tradition seems to have a legacy in Galileo, Discorsi . . ., p. 79 and Newton in a 1693 letter to Richard Bentley, in Cohen, Newton’s Papers and Letters on Natural Philosophy, pp. 293–299.

26 Henry of Harclay (MSS citt., note 4: Tortosa, 83v; Florence, 95r): “Dicitur quod infi nitum neque est maius neque minus, set maioritas vel minoritas est respectu ali-cuius fi nite. Contra: Ista proposicio est per se nota: ‘Omne totum est maius sua parte,’ et hec adhuc magis nota: ‘Illud quod continet aliud et aliquid ultra illud vel preter illud est totum respectu illius.’ Sic est in proposito. Nam totum tempus futurum ab hac die continet totum tempus futurum a crastina die et addit supra illud, igitur est totum respectu illius.” Reference to Grosseteste is occasioned by texts that say that, for example, the infi nite number of points contained in (but not, like Harclay, composed out of) a whole line is double the infi nite number contained in its half (see Comment. In VIII libros physicorum Aristotelis [ Dales], pp. 91–95 for Ockham, see Murdoch, “William of Ockham and the Logic of Infi nity and Continuity.” Gregory of Rimini, on the other hand, carries the analysis of parts and wholes and greater than and less than much further. Cf. Lectura super primum Sententiarum [ Trapp], dist. 42–44, q. 4, vol. 3, p. 458): In one way, he says, everything functions as a whole “which includes something and something else in addition to ( praeter) that something.” But in a second, more restricted way that is a whole “which includes something in the fi rst way and also includes a given amount more times than does that included (et includit tanta tot quot non includit inclusum).” An infi nite multitude can, Gregory continues, very well function as a whole with respect to another infi nite multitude, if “whole” is taken in sense one; but not in sense two. It seems clear, then, that what Gregory intends is, in our terms, a distinction between whole and part in the sense of set and subset (his fi rst meaning) and whole and part in the sense of unequal cardinality of the sets involved (his second meaning). Should there be any doubt, one need only look at the corresponding distinction he draws for the term “greater than”. In the strict sense (unequal cardinality) “a multitude is called greater which contains one more times or contains more units ( pluries continet

24 john e. murdoch

the infi nite, this alternative was most fruitful (although no medieval scholar, as far as I have been able to determine, took the additional step of defi ning the infi nite by means of the equality of part and whole; that was left to the nineteenth century).

Returning to the composition of continua, which was at least one of the considerations giving rise to the problem of unequal infi nites, we should note that, whether there were an infi nite number of indi-visibles or merely a fi nite number of them composing a continuum, they were, once again, inevitably extensionless. This is clear from the moves Aristotle made in Physics VI; but one thing he did not focus on was the existence of such extensionless indivisibles. Not so the fourteenth century. Indeed, almost all those embracing a nominalist ontology held that, in the strict sense, indivisibles did not exist. Ockham, for example, claimed that the term ‘point’ signifi es the same as ‘a line of such and such a length’ (linea tante vel tante longitudinis or linea non ulterius protensa vel extensa).27 Many others of nominalist persuasion say similar things.28 On the other hand, it has been objected to this nominalist defi nition of a point that, translating it into the geometrically true proposition that ‘between every two points there is always another point’ we would get ‘between every two lines of such and such a length there is always another line of such and such a length’, which would be, of course, absurd.29 Not so! It has to do with the defi nition of a point as a “line of such and such a length” and that is no more absurd than defi ning a

unum vel plures unitates)”; yet more generally (set/subset only), “every multitude which includes all the units of another multitude and certain other units is called greater than that (other multitude), even though it does not include more units than it (includit unitates omnes alterius multitudinis et quasdam alias unitates ab illis dicitur maior illa, esto quod non includat plures unitates quam illa).”

27 William of Ockham, Tractatus de quantitate et Tractatus de corpore Christi [Grassi ], OTh X, p. 22. Cf. William of Ockham, Expositio Physicorum [ Richter e.a.], III, ad tex. 71, OPh V, p. 585: “Unde non habent dicere quod punctus sit quoddam indivisibile distinctum a linea terminans ipsam lineam, sed debent dicere quod si sit res alia a linea, quod terminat lineam ex quo non potest esse sine linea et non est pars lineae.”

28 John Buridan, Questiones super libros Physicorum [ Paris, 1509], VI, 4, f. 97r–v: “Tunc ergo est dubitari quare punctum dicitur communiter ab omnibus esse indivisibile; respondetur quod hoc non dicitur quia sit ita vel quia sit verum de virtute sermonis, sed uno modo hoc dicitur secundum imaginationem mathematicorum ac si esset punctum indivisibile, non quia debeant credere quod ita sit, sed quia in mensurando revertuntur eedem conditiones sicut si ita esset.” Also, Thomas Bradwardine, Tractus de continuo (MS Torun, R.40.2, p. 192): “Superfi ciem, lineam sive punctum omnino non esse. Unde manifeste: Continuum non continuari nec fi nitari per talia, sed seipso.”

29 Kretzmann, Approaches to Nature in the Middle Ages, p. 215.


point as a pencil of lines (Veronese)30 or as an infi nite series of enclosure volumes (Whitehead and others).31 That is to say, once we (Ockham, Buridan, Veronese, Whitehead, etc.) have defi ned a point, then we can go on to speak of points in a normal way in their many occurrences in mathematics and natural philosophy. The medievals quite well realized this; such normal defi nitions of points, lines, surfaces, and instants in no way meant this had any effect upon points and the like in geometry and even upon arguments against indivisibilism or atomism of any sort.32

A case in point is at the beginning of Book VI of the Physics: Aristotle argues here that points or indivisibles have no parts and consequently they can only touch whole-to-whole; but if they do, there is no increase in size (non facit maius) of the continuous line they supposedly compose. This was one of the crucial arguments the late medieval atomists had to answer. Points or indivisibles had to have some connecting relation to one another to allow of faciens maius.

Thus, the indivisibilist Henry of Harclay says that points can very well cause an increase in size if they touch, or are applied to one another, secundum diversos situs.33 And Gerard of Odo, who is perhaps

30 Veronese, Fondamenti de geometria.31 Whitehead, An Enquiry Concerning the Principles of Natural Knowledge and The Concept

of Nature.32 For Ockham, Exp. Phys., III, ad text. 71 (ed. cit. & loc. cit.): “Et si aliquando auc-

tores ponant vocaliter talem propositionem categoricam, per eam intelligent unam ypotheticam. Nunc autem ad veritatem conditionalis non requiritur veritas anteceden-tis, et ideo ad mathematicas non requiritur quod aliquod infi nitum sit, sed requiritur quod ex tali propositione in qua ponitur iste terminus ‘infi nitum’ sequatur alia vel sequatur ex alia, et hoc potest contingere sine hoc quod infi nitum possit esse. Et sicut est de infi nito, ita est de puncto, linea et superfi cie. Et recte sensientes in mathematica et non transgredientes limites mathematice non asserunt quod punctus sit quaedam res indivisibilis distincta a linea nec linea a superfi cie nec superfi cies a corpore, sed ponunt conditionales aliquas in quibus subiicitur ‘punctus’ vel ‘linea’ vel ‘superfi cies’ sic accepta.” For Albert of Saxony, see his Sophismata [ Paris, 1495], unfol.; MS, Paris, BNF Lat. 16134, f. 43v): “. . . precisius loqui possumus imaginando instantia indivisibilia in tempore, licet talia in rei veritate non sint; nihilominus expedit ea imaginari . . . ita in proposito non plus neque minus dico quod expedit ea imaginari ad explicandum certas et precisas mensuras motuum et mutationum quas sine imaginatione instantium indivisibilium ita precise exprimere non possumus; nec ex hoc sequitur aliquod incon-veniens, quoniam sermones de talibus indivisibilibus per alias longas orationes debite exponuntur, propter quas etiam orationes prolixas evitandas expedit tales terminos ponere quos aliqui (ed. antiquos!) crediderunt supponere pro rebus veris indivisibilibus, licet tales res indivisibiles non sint nisi secundum imaginationem.” For Oresme, see his Questiones super libros physicorum (MS Sevilla, Colomb. 7–6–30, 67v): “Quod non est negandum indivisibilia esse, large et equivoce capiendo esse et ymaginando aliter quam mathematicus ymaginatur, quia talia sunt signifi cabilia.”

33 Henry of Harclay (MSS citt., note 4: Tortosa, f. 90r; Florence, f. 99r): “Ex istis

26 john e. murdoch

the most famous of the continental atomists, similarly claims that parts are distinguishable within indivisibles secundum differentias respectivas loci vel temporis, which are, for points, ante et retro, sursum et deorsum, dextrorsum et sinistrorum, and for instants, initium futuri et fi nis preteriti.34

On the other hand, in his opposition to indivisibilism Bradwardine grants the devil his due, as it were, and allows the respectable Euclidean notion of superpositio to function as the connecting relation between indivisibles.35 But then he establishes that any geometrical relation of superpositio is absolutely dissociated from continuity (or impositio, as he calls it).36

As an aside, it must be noted that the brilliance of Bradwardine’s geometrical and axiomatic treatment and continuity obscures the give and take of the actual arguments of the indivisibilists and their critics. Thus, though he opposes both Harclay and Chatton and cites them by name, it is diffi cult, if not impossible, to tell from his account what their detailed notions and arguments were.37

omnibus accipio quod necdum punctum, ymo nec linea nec corpus, facit maius extensive nisi applicetur secundum diversos situs. Ita dico quod duo indivisibilia, sicut puncta, si applicentur ad invicem secundum diversos situs, magis faciunt secundum situm.”

34 Gerard of Odo, Sent. I, dist. 37, MSS citt., note 3; Naples, ff. 238r–v; Valencia, ff. 22r–v) sets forth six deffensiva pro compositione continuorum ex indivisibilibus: “Nunc vero ponenda sunt quedam deffensiva pro opinione ista, que sunt sex in numero. Primum est indivisibile secundum partes est distinguibile et determinabile secundum differentias respectivas loci vel temporis. Istud declare in quinque generibus indivisibilium quan-titative. Primo in superfi cie que est indivisibilis secundum dimensionem profunditatis: Quoniam ipsa distinguitur et determinatur per intra et extra . . . Idem apparet, si suma-tur punctus in centro spere, et hoc secundum omnem differentiam localem: ante et retro, sursum et deorsum, dextrorssum et sinistrosum. Quod apparet, quia secundum differentiam circumvolvatur spera, pars que est sursum moveatur ante, pars deorsum movebitur retro, et sic de aliis oppositionibus.”

35 Superpositio also occurs in Averroes, Henry of Harclay and Gerard of Odo, but with a quite different, non-Euclidean, meaning. On all of this, see Murdoch, “Superposition, Congruence and Continuity in the Middle Ages.”

36 Thomas Bradwardine, Tractatus de continuo (MS Toruń R.40.2, pp. 158–160; Erfurt, Amploniana 40 385, ff. 19r–21r): “Conclusiones 8–13: 8. Inter nullas rectas sibi super-positas puncta alica mediare. 9. Lineam rectam secundum totum vel partem magnam recte alteri superponi et habere aliquod punctum intrinsecum commune cum ista non contingit. 10. Linee recte unam partem magnam alie recte imponi et aliam partem magnam superponi eidem vel ad latus distare ab illa impossibile comprobatur. 11. Unius recte duo puncta in alia continuari et per partem eius magnam superponi eidem vel ad latus distare ab illa non posse. 12. Linee recte unam partem magnam recte alteri superponi et aliam ad latus distare ab ista est impossibile manifestum. 13. Unius recte duo puncta alteri superponi vel unum imponi, aliud vero superponi et magnam eius partem ad latus distare ab ista non posse contingere.”

37 Thomas Bradwardine (MSS in previous note, p. 165, ff. 25v–26r): “Pro intellectu huius conclusionis est sciendum, quod circa compositionem continui sunt 5 opiniones


To continue to speak of the connecting relation between indivisibles, there is, however, another side to the late medieval atomists’ and their critics’ consideration of such a relation. This side involves the order of the constituent indivisibles within the continuum they compose. This question of order was not apparent in Aristotle or Bradwardine or in any of the relations between indivisibles à la Harclay or Odo. Order becomes involved when one asks of the relation of some one indivis-ible, not to a second indivisible, but to all other indivisibles (which are infi nite in number) in the continuum to which they belong. This is brought out neatly by the reply William of Alnwick gives to one of the positive arguments that Harclay provides for his belief that there are (an infi nite number of ) indivisibles which are immediate to one another. First the argument of Harclay:

For though my intellect does not understand how a continuum is com-posed of indivisibles that are immediately next to one another, the divine intellect necessarily does. One of my arguments for this view is the follow-ing: it is certain that God knows every point that can be designated in a continuum. Take, then, the fi rst inchoative point of a line; God perceives that point and any point in this line different from it. It follows, then, that either another line falls between the more immediate point He perceives or one does not. If not, then God perceives this point to be immediate to another one. If such a line does intercede, then, since points can be assigned in the line (which falls between the fi rst inchoative point and the other point), these mean points have not been perceived by God.38

famose inter veteres philosophos et modernos. Ponunt enim quidam, ut Aristoteles et Averroys et plurimi modernorum, continuum non componi ex athomis, sed ex partibus divisibilibus sine fi ne. Alii autem dicunt ipsum componi ex indivisibilibus dupliciter variantes, quoniam Democritus ponit continuum componi ex corporibus indivisibilibus. Alii autem ex punctis, et hii dupliciter, quia Pythagoras, pater huius secte, et Plato ac Waltherus modernus, ponunt ipsum componi ex fi nitis indivisibilibus. Alii autem ex infi nitis, et sunt bipartiti, quia quidam eorum, ut Henricus modernus, dicit ipsum componi ex infi nitis indivisibilibus immediate coniunctis; alii autem, ut Lyncul <nien-sis>, ex infi nitis ad invicem mediatis. Et ideo dicit conclusionem: ‘Si unum continuum componatur ex indivisibilibus secundum aliquem modum,’ intendendo per ‘modum’ aliquem predictorum modorum; tunc sequitur: ‘quodlibet continuum sic componi ex indivisibilibus secundum similem modum componendi’.”

38 Henry of Harclay (MSS citt., note 4: Tortosa, f. 88r; Florence, f. 98r): “Licet enim meus intellectus non comprehendit quomodo continuum componitur ex indivisibili-bus immediate se habentibus, tamen intellectus divinus hoc necessario comprehendit. Cuius est una racio hec: Certum est quod Deus modo intuetur omne punctum quod possit signari in continuo. Accipio igitur primum punctum in linea incoativum linee; Deus videt illum punctum et quodlibet aliud punctum ab isto in hac linea. Usque ad illum punctum inmediaciorem quem Deus videt intercipit alia linea aut non. Si non, Deus videt hunc punctum esse alteri inmediatum. Si sic, igitur cum in linea possint

28 john e. murdoch

Harclay’s argument stated in slightly different terms amounts to:

(0) God furnishes a manner of actualizing or specifying all the points in a given line (He is always conveniently on call to perform such tasks).

(1) God knows or perceives the initial point of the line and all others in the line.

(2) Consequently, God knows the relation of the initial point to all others.(3) If all such relations are of distance, then God does not know all oth-

ers, contra hypothesim.(4) Therefore, one such relation must be (not that of distance) but of

contact, which is to have indivisibles immediate to one another.

We then turn to William of Alnwick’s reply:

I reply in brief that this is true: (1) ‘between the fi rst point of the line and every other point of the same line known by God there is a mean line’. For any singular [of this universal ] is true, and, moreover, its contradic-tory is false. And this is so because the term ‘mean line’ in the predicate immediately following the universal sign [ i.e., ‘every’] has merely confused supposition. On the other hand, this is false: (2) ‘there is [some one] mean line between the fi rst point and every other point of the same line perceived by God’, since there is no [one] mean line between the fi rst point and every other point perceived by God. For there cannot be any such mean line, for if there were, it would fall between the fi rst point and itself; nor would that line be perceived by God. And therefore, when it is inferred: “if there is [such a mean line], then, as points can be assigned in the line, etc.,” the term ‘line’ there has particular supposition. And hence an inference is made affi rmatively from a superior to an inferior and thus the fallacy [of affi rming] the consequent is committed.39

signari puncta, illa puncta media non erant visa a Deo. Probacio huius consequencie: Nam per positum linea cadit inter hunc punctum primum et quodlibet aliud ab hoc puncto quod Deus videt; et ideo, per te modo inventum punctum medium Deus non videbat.” The term ‘immediaciorem’ is puzzling, since being immediate is not susceptible of degrees.

39 William of Alnwick, Determinatio 2 (MS Pal. lat. 1805, f. 14r–v): “Dico autem breviter quod ista est vera: ‘Inter primum punctum linee et omnem alium punctum eiusdem linee cognitum a Deo est linea media.’ Quelibet enim singularis est vera, et eius eciam contradictoria est falsa. Et hoc ideo est, quia ‘linea media’ in predicato sequens mediate signum universale stat confuse tantum. Hec tamen est falsa: ‘Est linea media inter primum punctum et omnem alium punctum eiusdem linee visum a Deo,’ quia nulla est linea media inter primum punctum et omnem alium punctum visum a Deo. Non enim contingit dare aliquam talem lineam mediam; sic enim mediaret inter primum punctum et seipsam, nec illa linea esset visa a Deo. Et ideo cum infertur: ‘Si


The crucial point in Alnwick’s reply is that there is a difference in supposition that accounts for the fact that proposition (1) is true while proposition (2) is false. Thus the term ‘mean line’ has merely confused (confuse tantum) supposition and, if we look in our “logical primer” about terms having this kind of supposition,40 we learn that no disjunctive descent can be made from such terms. That is, given proposition (1) ‘between the fi rst point of a line and every other point of the same line known by God, there is a mean line’, we cannot make a logical descent to the disjunction ‘either this mean line is between the fi rst point and every other point or that mean line is between the fi rst point and every other point or that other mean line is, etc.’ On the other hand, the false proposition (2), where the term ‘mean line’ has particular or determinate supposition, specifi cally allows such a disjunctive descent, causing the falsity of proposition (2) on grounds that any of the disjuncts is false (or to put it in Alnwick’s terms, there is, running disjunctively throughout the ‘mean lines’ involved, ‘no [one] mean line between the fi rst point and every other point’).

If, now, we translate what is being said in the medieval language of supposition into the notions of quantifi ers in modern logic, we can interpret the distinction between these two propositions as follows:

True (1) for all y there is an x such that x falls between the fi rst point and y.

False (2) there is an x such that for all y, x falls between the fi rst point and y.41

Here the universal and existential quantifi ers are reversed in the two propositions and we would say that the truth of (1) and the falsity of (2) derives from the fact that it is a case of multiple quantifi ers involving a

sic, igitur cum in linea possent puncta signari, et cetera,’ ibi ‘linea’ stat particulariter; et ideo arguitur a superiori ad inferius affi rmative et sic facit fallacia consequentis.”

40 For instance, Ockham’s Summa totius logicae, I, ch. 68, conveniently appearing in English translation with facing Latin text in his Philosophical Writings, ed. & tr. Philotheus Boehner, revised by Stephen Brown, pp. 70–74.

41 That is, in the standard notation: (y)( ∃x) 1<x<y( ∃x)(y) 1<x<y

Incidentally, Alnwick’s remark that “there cannot be any such mean line, for if there were, it would fall between the fi rst point and itself ” is quite correct, since it follows from proposition (2), without any further distinction in the scope of quantifi er, that ( ∃x) 1<x<x.

30 john e. murdoch

relational predicate (i.e., ‘mean’, ‘between’) where these quantifi ers are shifted in position. The medieval logician would say that the quantifi er shift has to do with a shift in supposition where the same term (‘mean line’) has a different supposition in proposition (1) and (2). Moreover, in any case, a term having merely confused supposition does not imply the same term with particular or determinate supposition.

Now the force of Alnwick’s criticism is that Harclay seems to be unaware of the fact that his argument blurs the distinction between propositions (1) and (2). Indeed, in Alnwick’s eyes, he would like proposi-tion (1) to imply proposition (2), which, we have seen, it does not.42

I have dealt more extensively with the question of the order of atoms or indivisibles within continua, using Harclay and Alnwick as a particularly good instance of what is at stake in this question. My reason for so doing is that this situation entailing a shift between sup-positions (or quantifi ers) is involved wherever ordered infi nite series, sequences, and processes are in question. And such question of order with respect to an infi nity of elements are met with everywhere in natural philosophy. Thus we fi nd a multiple quantifi cation or supposi-tion technique applied to Aristotle’s distinction between the potential and the actual infi nite,43 to the solution of Zeno’s “stadium” paradox,44

42 But since the (false) proposition (2) does imply the (true) proposition (1), then to argue from (1) to (2) means that Harclay has committed the fallacy of affi rming the consequent (as Alnwick says).

43 William of Ockham, Expositio physicorum, III text. 61, [ Wood e.a.] OPh V, p. 560: “Est autem istis adiciendum quod quamvis haec sit vera: ‘omni magnitudine est minor magnitudo,’ haec tamen est impossibilis: ‘aliqua magnitudo est minor omni magnitudine.’ Ista enim est vera: ‘omni magnitudine est minor magnitudo,’ quia est una universalis cuius quaelibet singularis est vera. Haec tamen est falsa: ‘aliqua magnitudo est minor omni magnitudine,’ quia est una particularis cuius quaelibet singularis est falsa. Et est simile sicut de istis duobus: haec est vera: ‘omnis homo est animal,’ et haec falsa: ‘aliquod animal est omnis homo.’ Et ratio diversitatis est quia in ista: ‘omni magnitudine est minor magnitudo; ly ‘minor magnitudo’ supponit confuse tantum propter signum universale praecedens a parte subiecti, et ideo ad veritatem suffi cit quod ista magnitudine sit una magnitudo minor et illa magnitudine sit una alia magnitudo minor et sic de aliis. Sed in ista: ‘aliqua magnitudo est minor omni magnitudine’ ly ‘magnitudo’ supponit determinate, et ideo oportet quod aliqua una magnitudo numero esset minor omni magnitudine, et per consequens esset minor seipsa.”

44 William of Ockham, Expositio physicorum, III, text. 79, [ Wood e.a.] OPh V, pp. 565–66: “Istis visis dicendum est quod intentio Philosophi est solvere rationes Zenonis per istum modum quod quia non sunt ibi partes infi nitae quarum quaelibet secundum se totam sit extra aliam, ita quod sit accipere primam, secundam et tertiam. Sed sunt ibi partes infi nitae quarum nulla est prima. Ideo non est inconveniens per illas partes infi nitas moveri. Unde ad primam rationem Zenonis, quando accipit quod impossibile


to the problem of the motion of a sphere over a plane surface (since it touches or is in contact with point after point in the plane),45 and other repetitions and refutations and arguments similar to that given by Henry of Harclay.46

What is more, in his criticism of Gerard of Odo’s atomism, John the Canon says that this technique amounts to a traditional rule and by means of its application recte possunt solvi multe rationes, which rationes all have to do with problems of infi nity or continuity.47 Of course this

est pertransire infi nita, verum est si illa infi nitae sint secundum se tota distincta, ita quod sit accipere aliquod de illis quo nullum sit prius. Sed talia non sunt in aliquo continuo. Si autem illa infi nita non sint secundum se tota distincta, ut non sit aliquod primum distinctum inter ea, possibile est talia infi nita pertransiri, immo necesse est talia infi nita pertransiri quandocumque aliquod spatium pertransitur.”

45 Walter Burley, Super Aristotelis libros de physica auscultatione commentaria [ Venice, 1589], col. 729: “Ad hanc formam dico, quod est concedenda, scilicet quod in plano continue est punctus post punctum, et tamen ista est falsa, scilicet quod in plano est punctus continue post punctum, quia nullus punctus est continue post alium punctum.”

46 Walter Burley, ibid., col. 730 (to give the essentials of a long text): “He concedes that sine medio aliud instans post instans A erit is true, but this does not imply the false proposition that aliud instans post A erit sine medio, since ‘sine medio’ (= ‘immediate’) is a syncategorematic term, and in the inference above would move from suppositio con-fuse tantum to suppositio determinate.” Gregory of Rimini, answering what is essentially Harclay’s argument (above, note 38), Sent. II, dist. 2, Q. 2 [Trapp], p. 292: “Ad tertium dico, stante praedicta suppositione, quod deus videt quod inter primum punctum et quodlibet aliud eiusdem lineae intercipitur linea, et infi nita etiam puncta, non tamen linea non visa, nec puncta non visa ab eo. Nullam tamen lineam deus videt intercipi inter primum punctum et quodlibet aliud punctum ab eo visum.”

47 John the Canon, Questiones in Physicorum VI, quaestio unica (MSS Florence, Bibl. Naz., conv. Soppr. C. 8 22 fol. 119v; Vat. Lat. 3013, fol. 73r): “Et pro quibusdam aliis rationibus solvendis: pro una, scilicet quod in continuo sunt plures partes quam infi nite, quia quelibet pars est divisibilis in infi nitum, applico istam regulam: Quod quandocunque arguitur ab alico termino communi supponente confuse tantum respectu alicuius magnitudinis ad eundem terminum supponentem personaliter respectu alicuius multitudinis, non est bona consequentia, quia sequitur fallacia fi gure dictionis. Quod declaratur in quadam consimili ratione. Solet enim probari a quibusdam quod multitudo non possit crescere in infi nitum, quia, si sic, tunc ultra omnem multitudinem fi nitam datam esset dare multitudinem fi nitam maiorem; sed multitudo maior omni multitudine fi nita est multitudo infi nita; ergo, si ultra omnem multitudinem fi nitam datam esset dare multitudinem fi nitam maiorem, sequeretur quod alica multitudo fi nita esset infi nita, quod est impossibile. Quod autem ultra omnem multitudinem fi nitam datam esset dare multitudinem fi nitam maiorem, si multitudo posset crescere in infi nitum, patet, quia quelibet singularis huius universalis foret vera, nam ultra hanc multitudinem fi nitam datam esset dare multitudinem fi nitam maiorem, et ultra illam et sic in infi nitum. Ad istam rationem respondetur quod hic est fallacia fi gure dictionis, quoniam in maiori iste terminus ‘multitudo’ in predicato prime propositionis supponit confuse tantum et dicit quale quid; in minori autem supponit tantum determinate respectu eiusdem multitudinis importate per signum universale; et ideo commutatur quale quid in hoc aliquid. Ita recte possunt solvi multe rationes iuxta istam materiam. Applicate, si vis.”

32 john e. murdoch

technique employing a difference in supposition is older than its newly found fourteenth-century application to such problems.48 Indeed, an improper understanding of what is involved in the technique is con-demned by Robert Kilwardby in 1277.49

Of course, questions of order of a different sort were in Aristotle himself when he asks of the permissible use of the indivisible begin-nings and endings of successive things (like motion) or permanent things changing within a continuous time (like something changing from white to not white).50 These passages in Aristotle led, as many of you know, to an enormous outburst of literature in the thirteenth- and particularly fourteenth-century, where some of it was both clever and fairly thought-provoking.51 This was the literature of “limit decisions” (as I and others have called it). Its concern was determining whether, to speak in modern terms, things were intrinsically or extrinsically limited (both successive things and permanent things against a continuous time).

Now one of the most creative aspects of this “limit decision” litera-ture was the many sophismata constructed to deal with these decisions. Often the function of sophisms was something akin to the following: one might be quite clear about what limit decision might be made in such and such a situation; but then a sophism might serve to reveal that things were not so clear after all and that the limit decision in question deserved further investigation.52

Further, the problem of limit decisions and the late medieval indivisi-bilists comes clearly into the picture when Bradwardine in his Tractatus de continuo showed that indivisibilism effectively destroyed the distinc-tion between intrinsic and extrinsic limits that was at the heart of all this fourteenth-century literature, something that would be preserved

48 For example, Peter of Spain, Tractatus, called afterwards Summule logicales [ De Rijk], pp. 222–23.

49 Chartularium Universitatis Parisiensis [ Denifl e e.a.] vol. I, p. 558: “Item quod non est suppositione in propositione magis pro supposito quam pro signifi cato, et ideo idem est dicere, cujuslibet hominis asinus currit, et asinus cujuslibet hominis currit.”

50 Physics, VI, ch. 5 and VIII, ch. 8, 263b9–264a6.51 For all of this, see Kretzmann, “Incipit/Desinit”; Murdoch, “Propositional Analysis

in Forteenth-Century Natural Philosophy: A Case Study”; above all, Wilson, William Heytesbury: Medieval Logic and the Rise of Mathematical Physics.

52 See, for example, The Sophismata of Richard Kilvington, both edited and translated by Norman and Barbara Kretzmann; Wilson (above, note 51), chs 2–3; Knuuttila, “Remarks on the Background of Fourtheenth Century Limit Decision Controversies”.


only if continua were infi nitely divisible.53 John Buridan also claimed a quite different inconsistency between maintaining atoms and properly expounding the beginning and ending limits of a thing. But he resolved this apparent inconsistency by making temporal intervals do the work that atomic instants were supposedly required to do.54

Bradwardine and Buridan were quite right in charging indivisibilists with, at least, obscuring limit decisions. For example, the indivisibilist Crathorn supported his contention that of no fi nite continuum is there an inifi nite number of proportional parts with a slightly humorous argument (which sounds as if it were drawn from some sophism-like literature on limits) that, if someone sins and then repents in succeeding alternate proportional parts of time, there is no way of knowing whether that person deserves damnation or salvation. Such a determination can only be made if there are a fi nite multitude of proportional parts.55

We do not have time or space to consider the late medieval atomists’ replies to all of Aristotle’s arguments against indivisibilism, let alone those of their critics in support of Aristotle. Yet I should mention one of the arguments in Physics, VI, ch. 2 about a mobile moving at various speeds over a given magnitude, that gave rise to considerable concern for the fourteenth-century atomist.

The centerpiece here in Aristotle was that, given a mobile moving over a given space or magnitude (s1) in a given time (t1), then (a) a faster mobile can of course move over the same magnitude in less time and (b) a slower mobile in the time of the faster mobile will move over less magnitude, and, therefore, (c) by alternately taking faster and slower

53 Thomas Bradwardine, Tractatus de continuo (MSS citt., note 36, p. 170, fol. 30r): “Concl. 50, Si sic, (scil. continuum compositum de indivisibilibus immediate conunctis) omni quod incipiet esse aliquale vel desinet esse tale secundum utramque signifi cationem incipiet vel desinet esse tale. Coroll. Cuiuslibet et qualiscumque rei talis esse primum intrinsecum et postremum.”

54 John Buridan, Quaestiones in libros Physicorum [ Paris, 1509], VI, q. 4: “Si autem aliquis vult aliter exponere incipit et desinit, ego dicam quod B incipit esse quia aliquo tempore est et immediate ante illud tempus non erat.” This was in line, of course, with Buridan’s nominalist removal of instants.

55 William Crathorn, Sent. I, q. 3, [ Hoffmann] p. 237: “Ponatur quod aliquis sic disponatur quod peccet in prima parte proportionali temporis et in secunda pae-niteat, iterate in tertia peccet et in quarta paeniteat et sic alternatim in aliis partibus proportionalibus temporis fi nite, cuius terminus sit a instans, in quo instanti volo quod moriatur. Isto casu posito aut salvabitur aut damnabitur. Neutrum potest dari, quia non est dare ultimum poenitentiae, quam non sequatur peccatum nec econverso. Igitur tali, qui infi nities peccavit et infi nities paenituit, non posset deus iusto iudicio aliquam poenam infl igere nec aliquod praemium conferre.”

34 john e. murdoch

mobiles, the faster will continuously divide the time, the slower the magnitude.56 Moreover, this argument assumes that (1) there can be faster and slower mobiles ad infi nitum, and (2) that magnitude, time and motion have a one-to-one correspondence between them (something that Aristotle had already said in his account of time in Book IV of the Physics).57

The atomists’ response to this was the following: Since they did not wish to give up mobiles moving faster and slower,58 they denied, in effect, one or the other, or both, of these two assumptions. Thus, some late medieval atomists maintained that there was a fastest motion (usually the motion of the primum mobile), thus denying the fi rst assumption.59 Similarly, the assumption is overturned by pointing out that fast and slow have to be determined by a different measure than moving in a given time more or less distance.60

56 Aristotle, Physics, VI, 2, 232a23–b20. Bradwardine supports what Aristotle says here in Concl. 24 of his Tractatus de continuo (MSS citt., note 36, p. 164, fol. 25r): “Quocumque motu locali signato potest motus localis uniformis et continuus omni proportione recte fi nite ad rectam fi nitam velocior et tardior inveniri. Coroll. Quodcumque spatium fi nitum quocumque tempore fi nito posse uniformiter et continue pertransiri.”

57 Aristotle, Physics, IV, 11, 219a10–14.58 For they did not maintain the alternative of the equality of speeds for atoms as

Epicurus’s had done (Diogene Laertius, X, 61) which, in any case, they did not have at their disposal. Nor were they cognizant of Epicurus’s answer to Aristotle’s argu-ment in Phys. VI, 2.

59 William Crathorn (ed. cit., note 55), pp. 242–43: “Nona conclusio est quod impos-sibile est aliquem motum esse velociorem motu vere continuo; licet enim unus motus vere unus et continuus, si aliquis talis sit, sit velocior motibus illis, qui non sunt vere continui, et motus non continuus sit velocior alio motu non continuo, tamen si aliquid vere continue moveatur non apparenter tantum, impossibile est aliquem motum tali motu esse velociorem.” He then gives three arguments in support of this conclusion. Nicholas of Autrecourt, Exigit ordo [O’Donnell], p. 215: “Sed illud latet nos ut supradixi, et tale mobile quod sic se habet quantum ad veritatem est mobile primum quod movetur motu velocissimo et de tali potest dici quod movetur in ipso nunc.” John Wyclif also maintained a fastest motion (Tractatus de logica, [ Dziewicki ] p. 39) where he said that “nichil potest velocius moveri motu successivo quam movetur equinoccialis”.

60 Gerard of Odo, Sent. I, dist. 37 (MSS citt., note 3) Naples, 239v, 241v; Valencia, 123r, 124r: “Probatio consequentie pro cuius evidentia premittuntur tres suppositiones. Prima est quod in omni tempore contingit aliquid moveri velocius et tardius. Secunda quod mobile velocius plus pertransit de spatio in equali tempore mobili tardiori. Tertia quod contingit mobili tardo duplicem sesquialteram seu emioliam longitudinem pertransiri a velociori . . . Ad probationem respondeo, primo ad suppositiones: Nego eas omnes tres ad intellectum ad quem inducuntur. Quando enim dicitur in prima quod in omni tempore contingit velocius et tardius moveri, si intelligatur quod in omni tempore contingit velocius et tardius moveri, hoc est, plus et minus pertransiri de spatio, suppositio est falsa simpliciter, quia possible est aliqua duo moveri, semper unum velocius altero, et numquam mobile velocius pertransibit plus de spatio quam mobile tardius. Unde si motus primi mobilis duraret per imperpetuum, polus articus


Yet another way was open to the medieval indivisibilists in accounting for the fast-slow mobile argument. It was to criticize the second assump-tion of the one-to-one correspondence between motion, magnitude and time. This assumption could be either according to one measure of taking time as continuous or by another measure of taking time as discrete. Thus, under time as continuous, two time intervals measuring motions contained an equal number of indivisibles when, and only when, the mobiles in question are moving over the same or equal magnitudes. However, if we consider time as discrete, then, if one time interval is less than, equal to, or greater than another time interval, then the infi nite number of indivisibles in the one is, correspondingly, less than, equal to, or greater than the other. Therefore, faster and slower are measured by discrete time, in effect answering Aristotle’s argument.61

Finally, there is a totally new element introduced into the debate about continuity. This consists in the consideration of the geometrical properties of the horn angle or angle of contingence (like DAC or EAC in Figure 2) or the angle of a semicircle (like BAD or BAE). The horn angle is one of the few geometrical objects which are intuitively infi nitesimal. The properties of these curvilinear angles were accurately expressed by Campanus of Novara in comments to his version of Euclid’s Elements.62 Thus, there are an infi nite number of horn angles (derivable by drawing greater and lesser circles through the same point of tangency) less than any acute rectilinear angle for example, DAC, EAC, etc., < FAC, and, mutatis mutandis, the same goes for angles of a semicircle relative to right angles. These are non-Archimedean magnitudes since the multiples of any horn angle will not exceed any rectilinear angle no matter how small. But Campanus noted that such angles do not obey the continuity principle: “if one moves through

et polus antarticus moverentur continue, non tamen plus pertransirent de spatio quam poli orbis lune, dato quod orbis lune non revolveretur nisi semel in centum annis et quod etiam continue moveretur.”

61 Henry of Harclay (MSS citt., note 4: Tortosa, fol. 93v; Florence, fol. 100v): “Et dico tunc breviter pro argumento quod accipiendo tempus et motum ut considerantur ut continua, non sunt plura instantia in duobus diebus quam in uno die, supposito quod equale spacium mensuretur per duos dies et per unum diem. Sed accipiendo tempus ut discretum, sic sunt plura instantia in duobus diebus quam in uno.” Chatton also criticizes the one-to-one correspondence assumption. For this, and for Harclay as well, see the discussion and texts in J. Murdoch, “Atomism and motion in the four-teenth century.”

62 In his comments to III, 15 (III, 16 of the Greek), where Euclid himself mentions a horn angle, and X, 1.

36 john e. murdoch

all mean values, from being less than something to being greater than it, then one moves through an equal.”63 Thus, given rectilinear angle BAF less than the semicircular angle BAE, by rotating AF about A toward AC, one will reach the right angle BAC moving through all acute rectilinear angles less than BAC, but never being equal to the semicircular angle BAE or for that matter BAD. In these observations Campanus was followed by any number of later medieval works, but notably by Bradwardine’s Geometria speculativa. Bradwardine also claimed that indivisibilism would mean that a straight line could divide a horn angle, contrary to what Euclid, and he himself, had maintained.64

Now these curvilinear angles entered into medieval philosophy and theology since it was maintained that they afforded a way to measure the infi nite distance or excess between quantities or things within one continuous latitude or series. Thus, already in 1290 Godfrey of Fontaines uses the relation between horn angles and rectilinear angles to explain how a fi nite caritas viae can be infi nitely exceeded by a fi nite caritas

63 This occurs in the course of Campanus comment to III, 15 [Basel, 1558]: “Hoc transit a minori ad maius, et per omnia media, ergo per aequale.”

64 Thomas Bradwardine, Geometria speculativa [ Molland], pp. 66–75. See also Bradwardine, Tractatus de continuo, concl. 7–8 (MSS citt., note 36: p. 176, fol. 36r): “Si sic (scil. continuum componatur ex indivisibilibus fi nitis), angulus contingentie dividetur per rectam.”

Fig. 2

B

DE

F

AC


patriae.65 Similarly, in the 1340’s Gregory of Rimini uses our angles to explain the infi nite distance between the fi nites albedo and nigredo.66

The champion, however, of applying curvilinear vs. rectilinear angles was Peter Ceffons, ca. 1348–49. Taking many of the assumptions of this application from Nicole Oresme,67 he turned to these angles to obtain a scale of measure that would prove to be effective when set against the different perfections of radically distinct species. He concludes that these angles do provide such a measure, fi lling out what he has in mind through no less than nineteen corollaries.68

To conclude this account of “angle calculus,” we might note that John of Ripa, who was no stranger in the application of mathematics to essentially theological problems,69 evidently thought that Peter Ceffons had gone too far in applying de proportionibus angulorum.70

This ends my catalogue or, at least, partial catalogue, of the ways in which in the fourteenth century quite new elements were brought to bear on Aristotle’s notions and arguments about infi nity and continuity. Some

65 Godefrey of Fontaines, Quodlibet VII, q. 12, [ De Wulf e.a.], p. 392: “Et ponitur exemplum de angulo recto et contingentiae; nam angulus contingentiae, quanto circulus est maior tanto angulus est maior (lege minor), sed nunquam tamen attingere potest ad hoc quod aequetur angulo recto (lege rectilineo), cum tamen quocumque circulo dato maior posset imaginari circulus et per consequens angulus contingentiae. Et hoc modo potest dici quod comparando caritatem viae ad caritatem patriae secundum hos modos perfectionum secundum suos actus, caritas viae posset augeri in infi nitum . . .” As can be seen, Godfrey’s, or the editor’s, mathematics is questionable in places.

66 Gregory of Rimini, Sent. I, dist. 17, q. 4 [ Trapp], vol. 2, pp. 410–411: “Ad pro-bationem, cum dicitur quod individuum albedinis, quod est A, excedit B individuum nigredinis sine proportione et per consequens in infi nitum, . . . simili modo posset probari quod quilibet angulus rectilineus esset infi nitus vel infi nitae magnitudinis. Sumatur enim unus rectus et sit A, et unus angulus contingentiae et sit B. Tunc probo quod A excedit B sine proportione.”

67 On his relation to Oresme, see now the fundamental article of Mazet, “Pierre Ceffons et Oresme—Leur relation revisitée”. It should be noted that Ceffons was some-thing of an inveterate “borrower,” often citing, in some cases verbatim, the likes of Thomas Bradwardine’s De proportionibus velocitatum, Roger Swineshead’s De obligationibus, John Mirecourt’s Comm. Sent., etc.

68 See Murdoch, “Mathesis in philosophiam scholasticam introducta: The Rise and Develop-ment of the Application of Mathematics in Fourteenth Century Philosophy and The-ology” and “Sublilitates Anglicanae in Fourteenth-Century Paris: John of Mirecourt and Peter Ceffons.”

69 John of Ripa, Quaestio de gradu supremo [Combes e.a.] and his Conclusiones (i.e. ex libris Sent. [Combes], pp. 70–72.

70 John of Ripa, Commentarius in Sententiarum, I, dist. 2, q. 4 (MS BNF, Lat. 15369, f. 147v): “Si arguantur contra premissas conclusiones per exempla mathematica, huius-modi exempla sunt refellenda, et maxime illa que ex proportionibus angulorum et ipso-rum excessibus arguunt consimiles proportiones et excessus inter species entium.”

38 john e. murdoch

of these elements grew from seeds already present in Aristotle’s text, such as his comments about faster and slower moving mobiles (whose whole purpose in these comments was to establish the infi nite divis-ibility of the continua involved), or such as the tremendous growth of literature in the fourteenth century of ever more complicated decisions regarding the limits of continuous processes and events. More strikingly new were elements entirely foreign to the Aristotelian seedbed including geometrical arguments against indivisibilism, the very question of the existence of such indivisibles, the consideration of the possibility of unequal infi nites, the differences of the logical doctrine of supposition as applied to the infi nite and the continuum and, lastly, the intrusion of curvilinear angles into debates about infi nite excesses.

INDIVISIBLES AND INFINITIES:RUFUS ON POINTS

Rega Wood

[ L]et us remember that we are dealing with infi nities and indivisibles, the former incomprehensible to our fi nite understanding by reason of their largeness, and the latter by their smallness. Yet we see that human reason does not want to abstain from giddying itself about them (Galileo, Discorsi e Dimostrazioni matematiche intorno a due nuoue scienze 1).1

Accounting for indivisibles consistently sometimes seems beyond even Aristotle’s capacity. His denial that indivisibles are quantitative, con-stitutive parts of the continua of magnitude, motion, and time is not completely consistent, despite the fundamental role this tenet plays in his physics. In De anima he cites uncritically the claim that straight touches sphere at a point,2 which suggests that points are integral parts of external bodies. In Physics 4.11, he sometimes describes instants not as temporal limits, but as units or numbers of time, and hence as constitutive parts, as Julia Annas has shown.3

As to their ontological status, Aristotle ridicules the suggestion that points or lines are substances in the Metaphysics (3.5.1002a24–b10). Yet, in the Posterior Analytics (1.27.87a36) he defi nes points as substance with position.

Whatever may be true about Aristotle himself on the relationship of mathematical objects to objects in the world, his interpreters are far from agreed. Some claim that the subject of Aristotle’s mathematics are ideal objects, not inhering in and quite independent of sensible

1 Galileo [ Leiden, 1638], p. 73. Translation: Two New Sciences [ Drake], p. 34.2 Aristotle, De anima 403a12–14.3 Annas, “Aristotle, Number and Time”. In the standard medieval Latin transla-

tion, these passages read as follows: Physica 4.11.4.12.220a3–4: “ipsum autem nunc est sicut id quod fertur, unitas est numeri;” 4.11.220b5–12: “tempus autem numerus est . . . quod numeratur, hic autem accidit prius et posterius semper alterum; ipsa enim nunc altera sunt. Est autem numerus unus quidem et idem qui est centum equorum et qui est centum hominum, quorum autem numerus est, altera sunt, equi ab homi-nibus”; 4.11.219b24–29 (Annas, pp. 11–12): “eo vero quod fertur cognoscimus prius et posterius in motu, secundum autem quod numerabile est prius et posterius, ipsum nunc . . . secundum enim quod numerabile est prius aut posterius, ipsum nunc est.” See Aristoteles Latinus 7.1.2, pp. 176–179.

40 rega wood

things. Jonathan Lear, by contrast, holds that Aristotle’s mathematics is about abstract objects only in so far as we consider mathematical objects in abstraction from their physical instantiation, and not in the sense that points and lines are not found in external objects.4 By contrast, thinkers like William Ockham have claimed that for Aristotle points were merely conceptual; that a point is nothing but a line not further extended, making it vain to defi ne a sense in which there are points in a line.5

If, pace Ockham, we suppose that at least for Aristotle, there are points in lines, indeed, potentially infi nitely many of them, it is not clear whether and how the infi nitely many points in one line can be com-pared to the infi nitely many points in lines of different lengths. Aristotle is clear that infi nities are not wholes, but in some sense parts (Physics 3.6.207a26–28; 3.7.208a14). And it is almost but not quite universally agreed that for Aristotle, all infi nities are equal, though he does not explicitly say so in Physics 3. 6–8,6 which is the passage normally cited for the view.7 Yet presumably, if pressed, Ockham, who claimed that one infi nity could be greater than another,8 would have credited Aristotle with this insight too. Other medieval authors who held that one infi n-ity could be greater than another include Robert Grosseteste, Henry Harclay, Adam Wodeham, and also Richard Rufus of Cornwall, author of the fi rst known Western commentary on Aristotle’s Metaphysics.

Given the complexity of Aristotle’s views on indivisibles, Rufus faced a formidable and complex interpretative challenge, which cannot be considered fully here. Instead I will consider Rufus’s response to only three questions:

1) Are points parts of external objects? 2) Are points and lines substances and parts of substances? 3) Are some infi nities of indivisibles greater than others?

4 Lear, “Aristotle’s Philosophy of Mathematics.”5 William of Ockham, Tractatus de quantitate [Grassi], OTh X, 1, p. 22; Expositio

Physicorum [ Wood e.a.], OPh V, 6.2, pp. 452–462.6 Thanks to Henry Mendell for confi rming that such an explicit statement is not

found elsewhere.7 See for example John Dorp, in John Buridan, Compendium totius logicae [ Venice,

1499; Frankfurt am Main, 1965], sign. 15 as cited by Ashworth, “An Early Fifteenth Century Discussion of Infi nite Sets,” pp. 232–233.

8 William of Ockham, Expositio in libros Physicorum [ Wood e.a.] OPh V, 6.6, p. 565; Quodlibeta septem [ Wey], OTh IX, 2.5, p. 132. For a discussion of Ockham’s views see Murdoch, “Ockham and the Logic of Infi nity and Continuity.”

indivisibles and infinities: rufus on points 41

1. Are there Points in the External World?

On the fi rst question Rufus and his contemporaries agree with Jonathan Lear. Points, like lines and spheres, are found in sensible objects, but mathematicians can consider them in abstraction from sensible objects without distortion.9 Typically this point is stated as a gloss on a text like Physics 2.2.193b22–25: “The next point to consider is how the mathematician differs from the student of nature; for natural bodies contain surfaces and volumes, lines and points, and these are the sub-ject matter of mathematics.”10 Rufus, like his contemporaries, defi nes mathematical abstraction by distinguishing it from physical and meta-physical abstraction.

Many such claims are commonplaces stated by a number of authors and in several of Rufus’s works; certain precisions occur, however, only in one work. So we will distinguish carefully positions stated only in a single work or group of works, since the attribution of Rufus’s commentaries on Physics (In Phys.) and De Anima (In DAn) has recently been challenged. Also considered here are two commentaries on Aristotle’s Metaphysics, Memoriale in Metaphysicam Aristotelis and Dissertatio in Metaphysicam (henceforth MMet and DMet) and two commentaries on Peter Lombard’s Sentences, Sententia Oxoniensis and Sententia Parisiensis (henceforth SOx and SPar).11

Firstly, I consider texts on how mathematical abstraction differs from physical abstraction and secondly texts which compare mathemati-cal abstraction with metaphysical abstraction. The following general conclusions will emerge. Mathematicians consider embodied objects, but do not consider them as such; by contrast metaphysicians con-sider disembodied or immaterial objects, objects apart from matter. Mathematical abstraction, unlike physical abstraction, prescinds from change; it considers its objects in abstraction from the sensible matter in which they inhere.

What variation is there among the accounts of the difference between physicist and mathematician and between physical and mathematical

9 Cf. Aristotle, Physics 2.2.193b34–35.10 Aristotle, Physics [ Barnes], I, p. 331.11 These work will be cited as follows MMet, Erfurt Quarto 290 (henceforth Q290);

DMet, Vat. lat. 4538 (henceforth V4538) and Q290; SOx, Balliol College 62 (henceforth B62), SPar, Vat. lat. 12993; In Phys. [ Wood], Oxford, 2003; In DAn, Madrid, Bibl. nac., 3314 and Erfurt Quarto 312 (henceforth M and Q312). For the controversy see Donati, “The anonymous commentary on the Physics.”

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objects? Unlike the other works, MMet distinguishes between primary and qualifi ed consideration and between primary and secondary sen-sible accidents. MMet tells us that the mathematician does not primarily consider sensible matter, but she does consider magnitude which con-cerns sensible matter. She abstracts from primary sensible accidents, the active and passive qualities that produce substantial change—namely, heat, cold, wet, and dry—but does not abstract from magnitude as a secondary quality.12

The Physics commentary tells us that the physicist defi nes things in terms of sensible matter; the mathematician, in terms of intelligible matter.13 Intelligible matter is a puzzling concept about which con-temporary Aristotelians do not agree.14 According to Lear, Aristotle invokes intelligible matter “to account for the fact that we are thinking about a particular object.” Perceptible objects “have intelligible matter insofar as they can be objects of thought rather than perception; that is, it is the object one is thinking about that has intelligible matter.” Albert the Great states a similar view, describing intelligible matter as conceptual or imaginable quantity.15 What Rufus believes is not entirely clear, but it seems he wants to defi ne intelligible matter negatively, and more generally, as unextended matter lacking position, and hence the appropriate proper subject for unextended accidents.16

12 MMet 6.2: “Dicendum quod mathematicus abstrahit a materia sensibili. Hoc est, non sic considerat materiam sensibilem sicut naturalis, quia non primo considerat materiam sensibilem; considerat tamen magnitudinem quae aliquo modo concernit materiam sensibilem . . . Abstrahit ergo a materia sensibili primo et immediate ut a corpore calido, frigido, humido, et sicco, et ita de aliis qualitatibus sensibilibus; non tamen abstrahit a magnitudine quae est sensibilis secundo” (Q290, f. 47vb). Cf. Aristotle, Metaphysics, 11.3. For a similar claim about magnitude as the ultimate subject of geometrical objects see Jones “Intelligible Matter and Geometry in Aristotle.”

13 In Phys. [ Wood ], 2.3.2, p. 121: “Dicendum quod materia dupliciter est: sensibilis et intelligibilis. Mathematicus tangit in defi nitione suorum accidentium materiam intel-ligibilem; naturalis, sensibilem materiam.” See also In Phys. P8, as quoted below.

14 Cf. Ross, Aristotle’s Metaphysics, II, pp. 199–200, Frede and Patzig, Aristoteles Metaphysik Z, II, pp. 195–196. Bostock in Aristotle, Metaphysics: Books Z and H, pp. 156–157, 284–285. Cf. Aristotle, Metaph. 7.10.1036a2–12, 7.11.1036b32–1037a5; 8.6.1045a33–36.

15 Lear, “Aristotle’s Philosophy of Mathematics,” p. 182. Albert the Great, Physica [Hossfeld] 1.1.1., Opera omnia 6.1: 2.

16 Rufus, DMet 8 (V4538, f. 65vb): “In aliquibus autem accidentibus, utpote in linea et in genere generalissimo lineae, ut in quantitate, et in aliis generibus generalissimis, est compositio ex essentia accidentis quae proprie dicitur esse et ex substantia materiae primae. Verbi gratia, linea dicit aggregationem ex essentia lineae et materia intelligibili. Unde linea dicit formam primo, et illa forma est illud quod addit linea super essentiam materiae primae situabilis”. Cf. Rufus, DMet 8 (V4538, f. 63vb): “Ad hoc dicendum quod sensibile unde sensibile situale est. Unde sensibile exigit materiam situalem, et


The claim that mathematicians do and physicists do not abstract from sensible matter is the single commonest statement on the subject we will encounter. The De anima commentary remarks that mathematicians consider sensible objects but not as such. And though mathematical objects can be abstracted from natural matter, here a synonym for sensible matter, they cannot be abstracted from intelligible matter. In abstracting from sensible matter, the mathematician prescinds from motion and change.17

DMet repeats the claim found in the Physics and De anima commentar-ies that though mathematical objects can be abstracted from sensible matter, they cannot be abstracted from intelligible matter. Unlike In Phys., DMet suggests that the defi nitions of the physicist and the mathe-matician can be the same, though the physicist’s defi nition explicitly references substance and the mathematician’s only implicitly.18

SOx tells us that natural objects differ from mathematical in that they include mobility.19 In his Parisian theology lectures, Rufus states many of the same distinctions listed above, though he expresses considerable skepticism about their signifi cance. Things which are transmutable are properly defi ned in terms of sensible matter and cannot be abstracted

ideo sensibile non est intelligibile ultimum, eo quod intelligibile ultimum situale non est. Hoc praedicatum igitur ‘habere materiam’—dico, situalem—defi nit sensibile. Prius enim est habere materialem situalem quam esse sensibile, et quod habet materiam situalem est sensibile, et ideo bene dicit cum dicit quod substantia sensibilis habet materiam, et debetis intelligere situalem; et per istam igitur propositionem innuit materiam esse”.

17 In DAn 1.1.Q1 (Q312, f. 19va): “Naturalia autem quamvis sint sensibilia et ita de facili apprehensibilia a nobis, causatur tamen in eis incertitudo. . . . Mathematicalia autem et sunt sensibilia (et ita facile apprehensibilia) et considerantur non ut cum motu et materia naturali consistunt, et propterea accipiunt ut bene nata perfi cere intellectum.”

In DAn 3.3.E3 (M, f. 81vb): “Hoc habito, quia ex iam dictis posset credi mathematica posse simpliciter abstrahi ab omni materia, incidenter subiungit quod sicut simum non potest abstrahi a materia naturali ut a naso, similiter rectum non potest separari a materia intelligibili, ut a continuo, et similiter de aliis mathematicis passionibus.”

18 DMet 1 (Q290, f. 4rb): “Mathematicus enim qui considerat de curvo si abstrahat a materia sensibili, non tamen a materia intelligibili.”

DMet 5 (Q290, f. 10vb): “Unde res mathematica quamvis abstrahatur a materia sensibili, non tamen a materia intelligibili.”

DMet 6 (Q290, f. 13rb): “Ad aliud dicendum quod defi nitione eadem contingere potest quod mathematicus et physicus utantur, sed non ut eadem. In defi nitione cuius-libet accidentis accipitur substantia, vel explicite vel implicite. In defi nitione autem mathematica, etsi accipiatur substantia, ipse tamen non percipit substantiam esse in illa defi nitione. Physicus autem percipit eam.”

19 SOx 2.13 (B62, f. 132ra): “Lux non est corpus mathematicum, quia lux natura est et addit super mathematicum, sicut linea radiosa super lineam mathematicam. Ergo si est corpus, est naturale. Sed omne corpus naturale est naturaliter mobile motu aliquo proprio naturali.”

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from sensible matter; mathematical objects cannot be abstracted from intelligible matter.

Sent. Par. 2.3: These things are said, but I do not see how they are adequate to the inquiry. I, too, can distinguish [concerning] matter: sensible matter differs from intelligible matter. Sensible matter is assumed and appears in defi nitions of transmutable things. Intelligible matter is also [assumed] for mathematical objects and they cannot be abstracted from this [intel-ligible] matter, as the Philosopher says.20

In general we are told that the objects studied by the mathematician can be considered in abstraction from sensible matter, not so those studied by the physicist. Mathematical objects can be considered in abstraction from the external objects in which they are embodied, since they are unconnected with motion and change. Since mathematical objects as such are not transmutable, they can be defi ned and abstracted from sensible matter, but they cannot be abstracted from intelligible matter. Some texts refer to natural and others to sensible objects, and this variation in terminology is found in the theological and metaphysical works as well as in the other Aristotle commentaries. Most different from the rest is MMet, in which there is no mention of that puzzling concept, intelligible matter.

How does mathematical differ from metaphysical abstraction? MMet tells us that the metaphysician abstracts from change, and hence since she considers objects “without sensible matter,” she studies their essences without change.21 In Phys., too, reserves to the metaphysician the treat-ment of things removed from motion and matter as such. By contrast mathematical objects are not actually separate, but are only considered separately. Though they are actually conjoined with matter, they are not considered as such.22

20 SPar 2.3 (V12993, f. 143va): “Dicta sunt haec, sed non video qualiter ad rem quaesitam satisfaciunt. Possum et ego distinguere materiam, nam alia est materia sensibilis, alia intelligibilis. Materia sensibilis accipitur et apparet in defi nitionibus rerum transmuta-bilium. Materia intelligibilis est etiam rerum mathematicarum et ab hac materia non abstrahuntur ut dicit Philosophus.”

21 MMet 6.2 (Q290, f. 48vb): “Sed nota quod metaphysicus considerat <consideras E> primas substantias sine materia sensibili, . . . et considerat eas essentias <pos. motu sed trp. E> sine motu.”

22 In Phys. [ Wood ], P8, pp. 90–91): “Res enim quaedam sunt secundum actum exsistendi et modum considerandi separatae a motu et materia; quaedam secundum actum exsistendi et modum considerandi coniunctae cum motu et materia; quaedam secundum actum exsistendi coniunctae, secundum modum considerandi separatae et abstractae. De primis est metaphysica tanquam de principali . . . De secundis rebus est


Neither In DAn nor DMet compare mathematician and metaphysi-cian. However they do compare the physicist or naturalist with the metaphysician. Specifi cally In De anima compares the treatment of the soul in psychology with that of the metaphysician. The psychologist is concerned with the soul as it actualizes a body; while the metaphysi-cian treats the soul as an absolute or separated substance. DMet makes the same claim on behalf of the metaphysician: His topic is not what actualizes a natural body. Rather, he considers the soul in so far as it is not in the body, the soul as spirit or intelligence.23

As mentioned earlier, Rufus’s take on mathematical abstraction is quite similar to Jonathan Lear’s, particularly in a couple of precisions found in In Phys. and In DAn, but at least apparently absent from DMet. Lear makes the point by saying that:

mathematical objects exist, but all this statement amounts to . . . is that mathematical properties are truly instantiated in physical objects and, by applying a predicate fi lter, we can consider these objects as solely instantiating the appropriate properties.24

The fi lter in question eliminates “predicates which concern the material composition of the object,” and allows the mathematician to consider only “the geometrical properties of objects and to posit objects that satisfy these properties alone.” Medievals would have described Lear’s fi lter, which he calls a “qua-operator,” in quite similar terms, using such expressions as ‘qua’ and ‘inquantum’—exempli gratia, ‘a body as a body’, or ‘a body as such’—reduplicatio, to use the medieval technical term.25

In the proem to In Phys. (paragraph 8), after telling us that though mathematical objects are conjoined with physical bodies, the

naturalis philosophia. De tertiis rebus est mathematica, sicut de magnitudine et numero. Ista enim licet secundum actum exsistendi sunt materiae coniuncta, sunt tamen secundum modum considerandi abstracta; considerantur enim non inquantum huiusmodi.”

23 Rufus, In DAn. 2 (Q312, f. 22va): “Haec enim differentia incorporeum est ipsius animae secundum quod anima est in se absoluta substantia, scilicet secundum quod anima est de consideratione metaphysici. Sic autem non intendit hic de anima sed secundum quod est natura sive actus corporis naturalis.”

Rufus, DMet 1 (V4538, f. 2rb): “Anima duplicem habet considerationem. Consideratur enim ipsa inquantum anima, et hoc est inquantum actus corporis naturalis, et sic etiam unita cum corpore. Consideratur autem alio modo inquantum spiritus vel intelligentia, et secundum istum modum non consideratur ipsa inquantum est in corpore. De ipsa autem considerata secundum primum modum est scientia de anima. De ipsa autem considerata secundum modum ultimum fi t perscrutatio in ista scientia.”

24 Lear, “Aristotle’s Philosophy of Mathematics,” p. 170.25 Ibid. pp. 168–169, 175.

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mathematician considers them abstractly, Rufus adds a further precision. He distinguishes between the negation of the mode and the mode of negation. Rufus says that:

In Phys. P9: In abstracting, he does not consider quality or number insofar as they are not connected with natural passions, but he considers them not insofar as they are such, as a reference to the negation of the mode and not the mode of negation.26

To suppose that number, points or other mathematical objects were not found in conjunction with natural passions would be to use the mode of negation—falsely, in this case, since in fact we encounter embod-ied spheres, for example. But to consider numbers as if they were not conjoined with natural passions would be to deny the [physical] mode, in order properly to understand the properties of spheres apart from bronze, iron or wood, for example.

Another way to put this is that the mathematician does not consider mathematical objects in so far as they are connected with physical objects, but equally does not deny that they are so connected. In the following snippet from the De anima citation quoted more fully in a previous note, this idea is expressed by the placement of the bolded ‘not ’, which indicates a “negation of the mode.”

In DAn 1.1.Q1: Mathematical objects are both sensible and considered not insofar as they are comprised with motion and natural matter, and accordingly they are taken in such a way as is well designed to perfect the intellect.27

DMet does not make this point in comparing the mathematician and the physicist. However in comparing the physicist and the metaphy-sician, it, too, makes the point by the placement of the ‘not’. Since metaphysician employs the mode of negation, rather than the negation of the [ physical] mode, the negation appears before the verb rather than before the description of the mode.

26 In Phys. [ Wood], P9, p. 92: “Non enim considerat abstrahens quantitatem vel numerum inquantum est non cum passionibus naturalibus, sed considerat non inquantum illius, ut tangatur negatio modi et non modum negationis.” The signifi cance of the technical terminology is a bit clearer in Scotus, where he uses it to state his understanding of the distinction between contraries and contradictories. See John Duns Scotus, Quaestiones super Praedicamenta Aristotelis [Andrews e.a.], Q. 39, p. 529.

27 In DAn 1.1.Q1 (Q312, f. 19va): “Mathematicalia autem et sunt sensibilia . . . et considerantur non ut cum motu et materia naturali consistunt, et propterea accipiuntur ut bene nata perfi cere intellectum.”


DMet 1: The soul is considered insofar as it is soul . . . united with the body. In another mode, however, it is considered as spirit or intelligence and according to this mode it is not considered insofar as it is in the body.28

In sum, by contrast to metaphysical abstraction, mathematical abstrac-tion is not about objects removed from the sensible world. Its objects are sensible and perceptible, not separated from physical objects, Since, however, they can be verbally and conceptually abstracted from physical objects, they are inseparable only from unsituated intelligible matter. But whether they are accidents inhering in intelligible or natural, situ-ated matter, the mathematician considers them without reference to the natural passions of change and corruption. The mathematician does not employ the mode of consideration proper to the natural, changing world, but neither does she posit independently existing, immaterial mathematical objects.

2. Are Points Substances or Parts of Substances?

This brings us to our second question: Are points substances, parts of substances, or accidents? The answer should be obvious, since points do not exist apart from bodies. But alas the answer is far from obvious, since Aristotle defi nes a point as a substance with position.29 Ordinarily, Rufus’s scrupulously refrains from citing this defi nition, though his contemporaries do not do so.30 Where he fi nds it diffi cult to avoid, he modifi es it, speaking not of point as substance with position, but of point being in substance in so far as it has position.31 One work, however, cites Aristotle’s defi nition: DMet 5, chapter 15, an exposition of the term ‘fi nal’ ( fi nis). Here Rufus’s dubs Aristotle’s defi nition the material defi nition and pairs it with a defi nition with which he is more comfortable, called the formal defi nition. As formally defi ned, point is a position of substance—an accident.

28 DMet 1 (V4538, f. 2rb): “Consideratur enim [anima] ipsa inquantum anima, et hoc est inquantum . . . unita cum corpore. Consideratur autem alio modo inquantum spiritus vel intelligentia, et secundum istum modum non consideratur ipsa inquantum est in corpore.” Quoted more fully above.

29 Posterior Analytics, 1.27.87a36. But cf. De anima 1.4.409a6.30 Cf. exempli gratia Anonymous Erfurt I, In De anima (Q312, f. 55rb): “Probatio

minoris: punctus, eo quod est substantia posita, est aliquid in se.” Robert Grosseteste, Commentarius in Posteriorum analyticorum libros [ Rossi ], 1.18, p. 258.

31 In DAn 1.4.Q2 (Q312, f. 20ra): “punctus autem est in substantia composita secun-dum quod habet positionem.”

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DMet 5.15: We should say that a point is substance in position when point is defi ned materially, but its formal defi nition is the position of substance.32

In chapter 1 on the term initial (initium), Rufus provides an unqualifi ed defi nition which is a variation of the formal defi nition: point is the fi rst disposition of matter when situated:

DMet 5.1: Hence a point is the fi rst disposition advening on matter when it is subject to site and it is like (quasi ) the origin of a line.33

But though strictly speaking a point is an accident, an accident of mat-ter, Rufus also wants to take seriously Aristotle’s claim that points are substances, substances in position. Similarly, lines are defi ned materi-ally as the substance of matter replicated infi nitely often in one direc-tion, while formally lines are defi ned as the replicability of matter in a single direction. If matter is broadly construed, a point is like (quasi) the origin of a line. This explains the sense in which a point is the cause of a line.

DMet 5.1: Point is the cause of line in the genus of material cause, tak-ing matter broadly.34

In the unqualifi ed defi nition of point as a quasi origin of the line, the language of mathematical construction, in which points are said to cause lines, is construed as a simile; a point is like the material cause. And the sense in which points and lines are substances is as they pertain to matter broadly construed. Like intelligible matter, matter broadly construed is an obscure concept. We know from DMet 7 that matter broadly construed is not subject to generation and corruption; it is an element in the composition of incorruptible ideas.35 Conceivably, Rufus is suggesting that we are in the realm of mental objects. If that sug-gestion were correct, then the claim might be that in mental geometry, points function as substances and materially cause lines.

32 DMet 5.15 (Q290, f. 10vb): “Ad aliud quod obicitur dicendum quod punctus est substantia posita, materialiter defi niendo punctum; formalis autem defi nitio est positio substantiae.”

33 DMet 5.1 (Q290, f. 9rb): “Unde punctus est prima dispositio adveniens materiae quando est sub situ et est quasi origo lineae.”

34 DMet 5 (V4538, f. 27va): “Ad aliud dicendum quod punctus est causa lineae in genere causae materialis, communiter sumendo materiam.”

35 DMet 7 (Q290, f. 20va): “Sed modo videtur quod haec consequentia quam sup-ponit Aristoteles—scilicet, quod si ideae sunt particulares, quod sunt corruptibiles—non teneat, sicut bene verum est quod ideae habent materiam communiter dictam, sed non materiam quod est subiectum generationis.”


However, the context of the claim that a point is the cause of a line shows that it is not a claim about the mental world, but a complicated metaphysical account of the external world. Points are the dispositions that give matter position and fi t it for extension and hence ultimately for corporality. It is as the fi rst disposition of matter in position that point is like (quasi ) the origin of a line. Similar accounts were offered by other authors. Richard Fishacre, for example, one of Rufus’s contemporaries, agrees that matter is disposed to receive the form of corporality by a form that gives it the position of a point, punctualis position. Fishacre would, however, reject Rufus’s defi nition of point as the fi rst disposition advening to matter, since he holds that unity is the fi rst disposition.36

Seeing matter broadly construed as matter prior to its disposition by points is an account that will serve for DMet 7, as well as DMet 5, of course. Prior to being disposed to assume position, matter is unextended and not subject of generation and corruption. Notice that Rufus’s claim that points are in some sense of the phrase ‘the material causes’ of lines does not commit him to the claim that points are constitutive parts of lines. Indeed, he makes the statement that if matter is broadly construed, points cause lines materially, shortly after denying that points make up a line.

DMet 5.1: [ W ]e should say that a point is the cause of a line in the genus of material cause, taking matter broadly. And he proves this as follows. In its material description a line is nothing but the substance of matter subject to position infi nitely often replicated. For matter subject to posi-tion once generates a point; infi nitely replicated, it generates a line. A line, therefore, is the substance of matter situated infi nitely often pointwise. Hence a line is generated from the substance of matter as it exists when subject to infi nitely many points.37

36 Richard Fishacre, Sent. [ Long, p. 9], 2.9: “Haec enim materia nunc sub forma corporali, si spoliaretur usque ad formam corporeitatis, et hac etiam adhuc forte est habens positionem et situalis et secundum imaginationem punctalis. Sed [si ] adhuc spolietur forma qua est situalis, et remanebit nondum omnino nudata. Habet enim adhuc formam primam, quae est tamquam unitas in formis, qua est una et sine posi-tione. Unitas enim est substantia sine positione.”

37 DMet 5.1 (Q290, f. 9rb): “Ad aliud dicendum quod punctus est causa lineae in genere causae materialis, communiter sumendo materiam. Et hoc declarat sic: linea enim nihil aliud est secundum materialem descriptionem nisi substantia materiae infi nities replicata sub situ; ipsa enim semel exsistens sub situ punctum gignit; infi nities replicata, lineam gignit. Linea igitur est substantia materiae infi nities situata punctualiter. Unde linea gignitur ex substantia materiae sub infi nitis punctis exsistens. Unde punctus est prima dispositio adveniens materiae quando est sub situ et est quasi origo lineae. Ex hoc potest dici, communiter loquendo, quod punctus est principium et causa lineae in genere causae materialis communiter acceptae.”

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When Rufus says that a line is the substance of matter situated point-wise ( punctualiter), presumably ‘pointwise’ explains how matter is situated in a line. If matter were not disposed by points it could not compose something with position. Since matter has no position prior to its disposition by points, there is a sense in which a point is the origin of any extended line.

Rufus’s mention of infi nite replicability would cause problems if he had suggested that points replicated infi nitely often made up the line, since that would imply that points were constitutive, integral parts. Since, however, it is matter as situated pointwise that is replicated, Rufus is committed neither to the claim that points are constitutive parts of the line nor to the claim that they can touch. That favors the account, since indivisibles cannot touch or be continuous with each other, both of which Aristotle defi nes as a relation of bodies with distinct limits (Phys. 6.1.231a21–22). Here, Robert Grosseteste appears to be the source of the discussion:

Physica 4: The replicability, therefore, of matter ad infi nitum is the principle of sensible things. For things having extension and sensible magnitude are not made from simple matter except by the infi nite replication of matter over itself.38

Notice that Grosseteste seems to say, but Rufus does not say, that simple matter is replicated, where simple matter presumably means unextended matter.

This is all a bit puzzling, and it would be nice to be clearer. But at least it should help us to understand how Rufus can in the same passage both affi rm that points are in some sense the material cause of lines and also deny that points are constitutive parts of lines. The denial appears just before the passage previously quoted, at the outset of the question

A similar statement is found in the chapter on ‘fi nis’. See DMet 5.15 (Q290, f. 10vb): “Ad aliud quod obicitur dicendum quod punctus est substantia posita, materialiter defi niendo punctum; formalis autem defi nitio est positio substantiae. Et similiter formalis defi nitio unitatis est non-positio substantiae. Et intelligatur ly ‘non’ privative. Linea autem secundum defi nitionem materialem est substantia materiae infi nities situaliter replicata secundum unam partem tantum. Defi nitio autem formalis est replicatio sive replicabilitas situaliter infi nita secundum unam partem tantum.”

38 Physica 3 [ Dales] p. 54: “Replicabilitas igitur materie in infi nitum numerus est et principium rerum sensibilium <add. ipsam replicacionem Dales>. De simplici namque <autem Dales> materia non fi erent res habentes extensionem et magnitudinem sen-sibilem, nisi per materie infi nitam super se replicacionem.” Corrections of the text courtesy of Neil Lewis.


Rufus asks about Metaph. 5.1.1012b34–1013a20, which begins: “We call an origin that part of a thing from which one would start fi rst, e.g. a line or a road has an origin in either of the contrary directions.” In the passage that concerns Rufus, Aristotle distinguishes between origins that are and are not immanent parts. As Rufus understands him, Aristotle posits two partial descriptions: ‘parts that make things’ and ‘parts that are in things’ and considers only two combinations: origins that make something but are outside it and origins that make something and are inside it. For the case of points in lines Rufus holds that we must allow a third possibility. Points are intrinsic parts that do not make up the things in which they are found:

DMet 5.1: We should say that ‘initial’ can be said in a third mode—namely, as that from which a thing is not made and it is in it . . . In another manner we should say that a point is the beginning (initium) of a line in the third mode, and a line is not made from points, but the substance of a line comes from them—that is, it rests on them (constat ex) and properly it is not made from them.39

Here, there is the puzzling affi rmation that lines rest on points, but are not made from them. I think we can make sense of the odd conjunction as a consequence of Rufus’s controversial claim that the substance of a line arises from points in that they dispose matter for position, though a line is not made from points.

Thus despite Rufus’s concession of an extended sense in which points are material causes of a line, it seems to me that Rufus escapes the dangers posed by Aristotle’s defi nition of points from the Posterior analytics, as well as the trap posed by positing a line generated from points. But the simile is common and alluring, given the indivisibil-ist mathematics of the day. In the Physics commentary, too, Rufus is tempted by it, suggesting that creation is like a line fl owing from the point which is our extensionless creator. But here too, Rufus escapes

39 DMet 5.1 (Q290, f. 9rb; V4538, f. 27va): “Item, cum ‘initium’ dicitur dupliciter: uno modo ex quo primo fi t res et est in ea, et alio modo dicitur initium ex quo fi t res et non est in ea, quare non habemus tertium modum ex quo non fi t res et est in ea? . . . Ad primam dicendum quod ‘initium’ potest dici tertio modo, scilicet ex quo non fi t res et est in ea. Et secundum hoc intelligit per illam litteram “ex quo fi t res prius et non est in ea.” [5.1.1013a7 Michael Scot tr.]. “Alio modo dicendum quod punctus est initium lineae secundum tertium modum et non fi t linea ex punctis, sed substantia lineae ex his, id est constat ex his et non fi t proprie ex his.”

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the problem, since the line fl ows rather than the point—in this case, the extrinsic limit of the fl owing line.40

Elsewhere in the Physics Rufus also rejects the claim that points are constitutive quantitive parts. He is no atomist; his denial is, if anything, more resolute than Aristotle’s. Commenting on Physics 6, he deals more consistently than Aristotle with the claim that a sphere touches a line at a point. Touching, for Rufus, is continuous motion or continuous varia-tion, and hence what happens at a point can only be part of any touch. He also tells us that rather than saying that contact is at indivisibile points, we should say that contact is “according to indivisibles.”

In Phys. 6.1.2: [ P]artial contact will be at a point. Yet not just points but lines touch. In this manner, the total contact that touches a line is nothing but a continuous variation by touching indivisibles. And just as we speak of the total contact in this manner, so we should say that in the contact what is touched in the whole contact is some continuum . . ., which is touched according to indivisibles. And just as the continuous variation by touching indivisibles is not touching indivisibles, . . . so it is true that a line is not its points.41

Note how Rufus carefully reminds us that though there are infi nitely many points in a line, points not only cannot touch, but do not con-stitute a line.

3. Are some Infinities of Indivisibles Greater than Others?

Specifying the manner in which something can be said precisely is typical of Rufus. And it has a major (if not infi nite) role to play in his solution to our third question: Are some some sets of infi nitely many indivisibles greater than others?

40 In Phys. [ Wood], 8.1.4, pp. 217–218.41 In Phys. 6.1.2 [ Wood] pp. 189–190: “Videtur quod oporteat ponere lineam esse ex

punctis hac ratione . . . Hoc mobile tetigit hanc lineam totam sed solum tetigit puncta. Ergo tota haec linea est puncta . . . Intelligendum est sic, quod sicut dictum est et pro-batum, quilibet contactus partialis erit in puncto. Nec tamen tanguntur solummodo puncta sed linea. Hoc modo totalis contactus quod tangit lineam nihil est nisi continua variatio per tangere indivisibilia. Et sicut dicimus hoc modo ex parte totalis contactus, ita debemus dicere ex parte contacti quod illud quod tangitur toto illo contactu est aliquid continuum . . . quod continue tangitur secundum puncta indivisibilia. Et sicut ipsa continua variatio secundum tangere indivisibilia non est ipsa tangere indivisibilia, sicut nec motus est ipsa mutata sed variatio secundum illa, sic verum est quod linea non est sua puncta”. See also ibid. 6.2.7, p. 199: “Sed si [punctum] moveretur per se, . . . essent tunc puncta partes magnitudinis et continua—quod est impossibile.”


Robert Grosseteste presented, perhaps for the fi rst time in the West, the claim that infi nities come in different sizes, arguing that some infi ni-ties contained others. Grosseteste claimed, for example, both that there were infi nitely many points in a one cubit line and that there were twice that number in a two cubit line. Indeed, he claimed that infi nities were related in every proportion, numerical and non numerical. Such mea-surement and comparison is impossible to us, to be sure, but possible to God for whom the infi nite is fi nite as Augustine teaches.42

Rufus fi rst quotes and responds to this position in a discussion of Metaphysics 10.1.1053a18–24,43 where he is considering Aristotle’s claim that we come to understand things by knowing their measure. Since infi nity destroys cognition, and continua are infi nitely divisible, Rufus asks how we can know them. He quotes Grosseteste claiming that though we cannot know the infi nitely many points in a line, God can. God, for whom the infi nite is fi nite, can compare infi nities and measure the greater by the lesser infi nite.

In response to Grosseteste, Rufus suggests an alternative approach: the proper measure of a line is potentially quantifi ed prime matter. In support of this claim Rufus repeats the claim he made in DMet 5 that a line is prime matter superimposed on by position infi nitely often. In a sense this is an endorsement of Grosseteste’s position, since it is Grosseteste who holds that replicating indivisibles infi nitely often produces fi nite quantity. It differs from Grosseteste chiefl y in not claim-ing that the matter so replicated is simple and in not invoking divine omnipotence explicitly.44

42 Grossesteste, Commentarius in VIII libros Physicorum [ Dales], 4, pp. 91–93.43 DMet 10 (Q290, f. 32ra): “Ad hoc respondet quis quod ista linea cognoscitur per

numerum infi nitorum punctorum in ipsa exsistentium. . . . apud ipsam [causam primam] est cognitio numeri punctorum infi nitorum.”

44 God may be invoked implicitly, however. Rufus had established to his satisfaction that matter, even prime matter, was intelligible. But though Rufus expressed no hesita-tion about our intellect’s capacity to grasp prime matter in Contra Averroem, a doubt is expressed in Speculum animae. On intelligibility, see Contra Averroem (Q312, f. 84vb): “De quaestione quarta et quinta quaerentibus de individuis et de substantia materiae primae, an sint simpliciter quantum est de se intelligibilia, an non, patet in tractatu illarum quod sunt intelligibilia, et in tractatu quintae et septimae quomodo sunt intel-ligibilia.” Regarding our intellects see CAv Ad 1 (Q312, f. 84rb): “omne ens et natura essentialiter est intelligibile ab intellectu primo, similiter <simpliciter E> autem et quantum est de se ab intellectu causato.”

Compare Speculum animae 4 (Q312, f. 109va): “nam omnis <rep. E> creatura [est] vere <animae A> intelligibilis, et ab intellectu Primo et ab intellectu creato, nisi sit defectus a parte nostri intellectus, propter quem scilicet non possit ipsa principia prima intelligere—quae principia quantum in se ipsis est <!> maxime intelligibilia sunt.”

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DMet 10: Prime matter is the origin of all quantities. Hence it is the origin of every dimension of continuous quantity whatever. For when posited under a single position, it makes a point, but when replicated under position in a single direction infi nitely often, it makes a line. . . . Thus matter as a potentially quantifi ed entity is the proper measure of any quantity whatever; as a potentially quantifi ed entity in one dimension it is the measure of a line. Moreover so disposed it pertains to the same category as that which it measures.45

Here, it looks rather as if Rufus is trying to explain how something dimensionless can measure dimensions. Prime matter is one and unex-tended, lacking any determinate dimensions, but when it is disposed for position, it is potentially extended and divisible. The suggestion seems to be that when prime matter’s potential for quantity is actualized in infi nitely many different positions, this results in extension. This seems a somewhat uncritical response to Grosseteste.

Fortunately, therefore, in his Oxford lectures Rufus has a second look. This time he is both considerably more critical and at the same time seemingly more committed to one of Grosseteste’s most controversial views. Rufus explicitly endorses Grosseteste’s claim that the infi nity of points in a short line is less than the infi nity of points in a longer line.

Let us start our examination of the Oxford discussion by looking at the dilemma as Rufus presents it. It seems that the numbers succeeding 10 are fewer than the numbers succeeding 1. And nothing prevents indeterminate quantities from being greater than or lesser than each

45 DMet 10 (Q290, f. 32ra): “Sed alio modo potest responderi quod materia prima est origo omnium quantitatum <qualitatum E>. Unde ipsa est origo cuiuslibet dimensionis quantitatis continuae. Ipsa enim sub uno situ posita producit punctum; ipsa autem infi ni-ties replicata sub situ in unam partem est causa lineae. Ipsa autem infi nities replicata secundum duas dimensiones producit superfi ciem. Ipsa autem secundum triplicem dimensionem producit corpus. Sic igitur materia [ut] ens in potentia quanta est men-sura propria cuiuslibet quantitatis, ut ipsum ens in potentia quanta secundum unam dimensionem est mensura <mensura lineae] mensurabile E> lineae; ipsa [autem] sic disposita est eiusdem praedicamenti <puncti E> cum eo <eius E> cuius est mensura, et ita <illa E> mensura et mensuratum sunt unigenea.”

For a similar discussion see MMet 10.5 (Q290, f. 52ra): “Ad quod dicendum sine oppositione quod materia, ens in potentia quanta, est unum et minimum et mensura rerum in quantitate exsistentium, sed haec diversifi catur per differentias reales. Prout enim est considerata sub situ sic est subiectum puncti; prout vero non sub situ, sic uni-tatis. Haec eadem iterum materia, <add. non E> prout sub situ considerata, infi nities replicata secundum unam extensionem gignit lineam; secundum vero duas, superfi ciem; secundum vero tres, corpus. Et sic patet qualiter materia est subiectum puncti, unitatis, numeri, lineae, superfi ciei et corporis.”


other; one not precisely measurable pile can be greater than another. So it seems that one infi nity can be greater than another. Rufus asks:

SOx 1.2H: [ I ]n one line there are infi nitely many lines, and in half of it there are infi nitely many lines, and is this infi nity more than that? Again, numbers increase infi nitely from 10; they also increase from one. Is this infi nity greater than that? Is that infi nite greater than this by the ten added to it? Again, the account (ratio) of fi nite and infi nite pertains to quantity. Great, small, greater, lesser are indeterminate quantities, and none of them confl icts with the account (ratio) of the infi nite, therefore nothing prohibits one infi nity’s being greater than another.46

Next come quotations of Grosseteste and Augustine supporting this claim, followed by what looks like a rejection of these authors. They do not understand the account of infi nity; they seem to have confused the account of infi nity with the account of all, as in God sees all things. But to see all is to see a whole; so if God sees all, what he sees is fi nite not infi nite, since an infi nite is not a whole but incomplete.

SOx 1.2H: These [thinkers] do not appear correctly to comprehend the account (ratio) of infi nity. For “beyond which there is nothing” is not the account of infi nity, but rather this is the account of that which is all; however, all and whole and perfect are the same [account,47 and] hence ‘all’ indicates the fi nite. Therefore, the foregoing seems rather to be the account of the fi nite than the infi nite.48

Rufus then explains that ‘what contains everything within itself ’ is not a correct description of the infi nite, rather something is infi nite if whatever quantity we assign to it, there will be a further quantity. To be beyond an infi nite [series] is impossible, since such a series is unending. And therefore, we expect Rufus to conclude that one infi nity cannot exceed another. Though he does not state it explicitly, Rufus is

46 SOx 1.2H (B62, f. 22rb): “Sed contra hoc videtur: in una linea sunt infi nitae lineae, et in eius medietate sunt infi nitae lineae. Et nonne haec infi nita sunt plura illis infi nitis? Item, ascendat numerus in infi nitum a denario, ascendat etiam et ab unitate. Nonne istud infi nitum est maius illo? Nonne habet decem unitates additas illi? Item, quantitati congruit ratio fi niti et infi niti. Magnum, parvum, maius, minus sunt quanti-tates et indeterminatae, ergo neutri illorum repugnat ratio infi niti, ergo nihil prohibet infi nitum esse maius infi nito.”

47 Cf. Aristot., De Caelo 1.6.268a20.48 SOx 1.2H (B62, f. 22rb): “Non videntur isti recte accipere rationem infi niti. Non

est enim haec ratio infi niti ‘extra quod nihil’. Sed est haec potius ratio eius quod est omne; omne autem et totum et perfectum idem, quare ‘omne’ fi nitum dicit. Ergo praedicta potius est ratio fi niti quam infi niti.”

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committed to this conclusion. He holds that exceeding or being outside (extra) the infi nite is a contradiction in terms.

SOx 1.2H: Again, according to the Philosopher [ Phys. 3.6.207a7–8] what contains everything within itself is not the account of the infi nite, but rather “something is infi nite, if [whatever] we take as its quantity there is always something further (extra).” It is pointless to say that ‘the infi nite is that outside (extra) which there is something’, for ‘outside’ fi nishes and terminates, hence it confl icts with (repugnat) the infi nite, and the phrase ‘beyond (extra) the infi nite’ is a contradiction in terms.49

In this respect Rufus’s solution is preferable to Ockham’s, since Ockham, at least when he is not expressing himself carefully, allows that one infi nity exceeds another.50

Harder to understand is the concession Rufus makes to Grosseteste: one infi nite can be greater than another, though one infi nite cannot exceed another,

SOx 1.2H: Again, the infi nite is not a whole, the infi nite is not a part, and yet an infi nite is greater than an infi nite.51

Rufus explains his concession by reference to a simile found in De anima. The common sense as it judges sensible species from different senses is described as a point using the termini of two paths: 3.2.427a10–15. Rufus compares the common sense to a point at the center of a circle that is numerically one by substance and subject, yet infi nitely many in being and account (rationem).

SOx 1.2H: Again, in a circle there is a point at the center, and it is numeri-cally one in its subject, yet in being and account (ratio) it is as multiple and as many as the lines terminated at it are many (Cf. DAn 3.2.427a9–14). Therefore the point itself is (as it were) infi nitely many points, though only in

49 SOx 1.2H (B62, f. 22rb): “Item, non est haec ratio infi niti quod continet in se omne, sed haec est ratio infi niti, secundum Philosophum: ‘Infi nitum est cuius quanti-tatem accipientibus semper est aliquid extra sumere’. Nec est aliquid dictu ‘infi nitum est extra quod est aliquid’; nam ‘extra’ fi nit et terminat. Unde opponitur infi nito, et est oppositio in adiecto ‘extra infi nitum’.”

50 William of Ockham, Quaestiones variae 1 [ Etzkorn & Kelley], OTh VIII, p. 80: “Tamen revolutiones lunae sunt plures infi nitates quam revolutiones solis. Et ideo posita hypothesis debet concedi quod infi nitum est maius infi nito et exceditur ab infi nito.” More cau-tiously in Quodlibet 2.5, Ockham claims “tot sunt ista et adhuc sunt multa.” However, he still does not absolutely deny that one infi nity can exceed another, just that such an excess can be determinate (certo numero).

51 SOx 1.2H (B62, f. 22rb): “Item, infi nitum non est totum, infi nitum non est pars (cf. Physica, 3.6.207a26–28), et tamen infi nitum est maius infi nito.”


its being and account, and yet by substance and subject it is single (unicus). And therefore it is not true that a single point would contain as many points as a maximal line or as many as the world machine.52

Rufus tells us that points can be numbered either by substance or account. There are not as many points at the center of the circle as there are in a maximal line. But he does not tell us whether this is because there is only one substantial point at the center of a circle, or because the being or account (ratio) of the point at the center of the circle can be counted in fewer ways than the points in a maximal line. Going from a circle to a straight line, Rufus again concludes that since single substantial points can be many in being, there are more in longer than in shorter lines.

SOx 1.2H: Points can be numbered in two ways, as is already evident, namely either by substance and subject or by being. At the extremities of a single line there are two points, two in reality (rem) and subject, but in the line itself any one point is one by number and subject, [ yet] twofold by being and account (rationem), in that it is the beginning of one line and the end of another. In this manner (modum), there are fewer points in the shorter line and more in the longer, yet there are infi nitely many in both.53

Since this conclusion, and in particular the phrase “in this manner,” is hard to understand, we should look at Rufus’s De anima commentary for further clarifi cation. Of the six early De anima commentaries I know (and not many more survive) it is the only one which explains Aristotle’s example in terms of the distinction between the substance of points and their account (ratio) and being, the distinction that is key to Rufus’s Oxford explanation of how one infi nity can be greater than another.54

52 SOx 1.2H (B62, f.22rb): “Item, in circulo est punctus qui est centrum, et est unicus numero secundum subiectum, tam multiplex tamen sive tam multi secundum esse et rationem quam multae sunt lineae ad ipsum terminatae. Est igitur ipse punctus quasi infi niti puncti, sed solum secundum esse et rationem, tamen substantia et subiecto unicus est. Et ideo non est verum quod unicus punctus contineat tot punctos quot et maxima linea sive quot et mundi machina.”

53 SOx 1.2H (B62, f.22rb): “Dupliciter enim est numerare punctos, ut iam patet, scilicet secundum substantiam et subiectum, aut secundum esse. In extremitatibus unius lineae duo puncti sunt, duo secundum rem et subiectum, in ipsa vero linea quivis unus punctus unus est numero et subiecto, duplex secundum esse et rationem, eo quod principium est unius lineae et fi nis alterius. Secundum hunc modum sunt in breviore linea pauciores puncti et in longiore plures, infi niti tamen in utraque.”

54 Others consulted use somewhat different terminology, sometimes not naming the sense in which the point is many. See Anonymous, Sententia super II et III De anima [Bazán], 2.26, pp. 343–347: essence and being (hereafter, Anonymous Bodley). Anonymous Erfurt II, In Dan 2, Q312.65va; subject. Anonymous, Lectura in librum de anima a quodam discipulo

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Aristotle, and Rufus following him, employ the analogy to explain how the common sense can simultaneously be aware of sensibles received by two distinct senses—yellow and sweet, for example.

In DAn 2.12.E5: But the common sense is not in [ just] any matter what-ever one and diverse by account (rationem), but it is one and indivisible as it is in itself one and indivisible; and because it is indivisible in this manner, it is one discerning [subject] at the same time. And just as the same point as a whole [and ] in itself pertains in account (ratione) to diverse things in so far as it is the terminus of diverse lines meeting (concurrentium), so similarly the common sense, since it is one indivisible by substance, as a whole [and] in itself pertains in account to diverse things in so far as it is the terminus of two diverse paths leading from the particular senses. And in this manner its being does not pertain to one indivisible by account, since the soul uses one indivisible by substance twice at the same time, in so far as it is the terminus of two paths leading from some two senses. And so in this way it uses the same terminus according to substance as two, namely [directed] toward two paths of two senses, [and] it judges the diverse sensibles of two senses. In so far as it is one by substance it judges these at one and the same time.55

Controversially, Rufus claims that because the common sense is indivis-ible and unextended, it can simultaneously receive two distinct sensibles, unlike an extended body which cannot at the same time be white and black.56 Here we may have switched from the common sense to the sense of vision, but that will not matter for Rufus’ claim: though a

reportata [Gauthier], 2.25, p. 419: substance and apprehension. Adam Buckfi eld, In DAn 2, [ Powell ], p. 190: subject; grateful thanks to Miss Powell for permission to cite Adam. Ps. Buckfi eld, In DAn 2, Merton College 272.17va, subject or substance.

55 In DAn 2.12.E5 (M3314, ff. 78vb–79ra): “Consequenter solvit hanc rationem dicens: Sed sensus communis non est quocumquemodo unum et secundum rationem diversum, sed est unum et indivisibile sicut punctus <penitus M> in se est unus et indivisibilis; et quia sic est indivisibile, est unum discernens et in eodem tempore. Et sicut idem punctus est secundum se totum in ratione diversorum, prout est terminus diversarum linearum concurrentium, similiter sensus communis cum sit unum indivisi-bile secundum substantiam, est secundum se totum in ratione diversorum secundum quod est terminus diversarum duarum viarum ductarum a sensibus particularibus. Et secundum hoc esse eius non est in ratione unius indivisibilis, quia anima utitur uno indivisibili secundum substantiam in eodem tempore bis, secundum quod est terminus duarum viarum a duobus aliquibus sensibus ductarum. Et secundum quod sic utitur termino eodem secundum substantiam tamquam duobus, scilicet ad duas vias duorum sensuum, sic duorum sensuum sensibilia diversa iudicat. Inquantum autem est unum secundum substantiam, sic uno eodemque tempore haec iudicat.”

56 Anonymous Bodley, In DAn 2.26, takes this to be Aristotle expounding an opinion with which he disagrees. According to Anonymous Bodley the common sense is entirely indivisible, and it is impossible that it should be divisible in accordance with the diverse sensibles it apprehends (ed. Bazán, pp. 343–347).


body cannot be both white and black at the same time (and in the same respect), a point can. Because unextended entities as a whole can pertain in account to diverse things, we can simultaneously sense black and white.

In DAn 2.12.Q2: And by his saying that the common sense is one and diverse, as is a point (3.2.427a9–12), we should understand that he solves the aforesaid doubt suffi ciently. For in this way it is evident that the com-mon sense and a body are not similar. For if it were possible that some body as whole in itself should have diverse accounts at once, so that it could by one account receive whiteness and according to another account blackness, it would be possible for the same body to be white and black at once. But now that is not possible, but the common sense can be so, pertaining in account to diverse things at once by itself as a whole. . . .57

When Rufus asserts that because our senses are unextended as points are, contrary qualities can simultaneously be predicated of them, he cannot be referring to something conceptual. When he speaks about numbering points according to their capacity for different accounts, what is differently numbered will pertain to the external world. In the case of two spheres colored white and black, for example, the part of their touch that occurs at a point is at a point that is simultaneously the limit of something white and the limit of something black, but uncolored because unextended.

Just how this solves the problem of how one infi nity can be greater than, without being able to exceed another, is still not clear, however. Here is the fi rst of three alternatives: Perhaps Rufus is suggesting that there will be more points that both begin and end a one inch line in a four inch than in a three inch line, more connecting points about which we can claim that they are the beginning and end of one inch lines.

57 In DAn 2.12.Q2 (M3314, f. 79ra): “Et intelligendum quod per hoc quod dixit sensum communem esse unum <idem M> et diversum sicut punctus [3.2.427a9–12], suffi cienter dissolvit dubitationem praedictam. Quia <Et M> per hoc patet quod non est simile de sensu <sensi M> communi et de corpore; si enim esset possibile aliquod corpus secundum se totum simul se habere secundum diversas rationes, ita quod possit secundum unam rationem recipere albedinem et secundum aliam rationem nigredinem, esset possibile idem corpus simul esse album et nigrum. Nunc autem non est illud pos-sibile, sed sensus communis <om. M> potest esse sic in ratione diversorum secundum se totum simul. Intelligatur enim locus quidam in corpore in quo est situm instrumentum ipsius sensus communis a quo instrumento exeunt et extenduntur venae sive viae diversae ad organa singulorum sensuum. Et in isto <add. sed exp. ig M> organo in quo concurrunt huiusmodi viae <corr. (s. lin.) ex: venae M> radicatur sensus communis, qui manens unus et idem secundum substantiam secundum quod ad ipsius organum terminatur via ducta ab organo visus, secundum se totum immutatur et recipit visibile.”

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That is to say in a two inch line, there is only one point that connects two one inch lines. In a three inch line, there will be potentially infi nitely many such points, since every point between the one inch and the two inch mark will do so. In a four inch line, there will be even more such points, since there will be infi nitely many such points between the two inch mark and the three inch mark, as well as between the one inch mark and the two inch mark.

If this is correct, Rufus’s notion of numbering by being or account will result in claims for the differential greatness of various infi nities that are parasitic on the different fi nite quantities associated with them, their included parts. Nonetheless, it is the kind of thing that might make a difference. So far we have looked at intensive infi nities that arise in an unending process of division. Now consider an extensive infi nity, produced by a process of addition or multiplication. The example proposed by Nicholas Denyer is an infi nitely long prison sen-tence. Given the choice of two such sentences, each of which is for an unlimited number of years, we prefer a sentence of one day a year in hell rather than a sentence of 364 days a year,58 since any fi nite part of that sentence includes more days. Neither sentence extends beyond the other, however. Indeed, since the two sentences can be put in one-to-one correspondence, they have the same cardinality and hence are equal. But since the one is a proper subset of the other, there is also a sense in which one is greater than the other.59 The fi rst interpretation credits Rufus with insight into this second sense of greatness. In the case that concerns him, neither line can be further divided than the other, but there will be more of given size line lengths in the longer than the shorter line, and hence the number of points corresponding to the connecting points in the shorter line will be a subset of the number of connecting points in the longer line.

A second interpretation is possible, and it seems more likely. Rufus may not have anything in mind like aliquot parts that divide a line without remainder, since he does not mention such parts. In fact both in his Oxford lectures and in his De anima commentary, he speaks of points that are twofold in account (rationem). This suggests that he is counting and comparing two infi nities of points with actual common end points, and perhaps in comparing one to another, superposition

58 Sorabji, Time Creation and the Continuum, p. 218.59 Maddy, “Proper Classes”, p. 114.


plays an important role. Perhaps Rufus is thinking of comparing two unequal lines, each of which is composed of infi nitely many points: call them AC (shorter) and AD (longer).60 Intermediate points in both lines will be both the end of lines beginning at the left and the beginning of lines ending at the right (or vice versa). If, however, the shorter is superimposed on the longer, intermediate points in the longer line (AD) overlapped by the shorter line (AC) will begin both lines that end where AC ends and lines that end where AD ends. Since those points can be counted as the termini of more lines, there will be more accounts of points in AD. Since we can count them as connecting lines to more actual end points, points in AD will be greater in account than points in AC. And the same will hold for any superimposed or superimpos-able infi nities. I suppose that Rufus considers only lines with actual end points, because he refers to the intermediate points as the beginning and end of only two lines.

As in the previous interpretation, the sense in which one infi nite is greater than another depends on their fi nite relationship. However the sense in which one is greater than the other will differ; and there is no sense in which the relationship could be considered proportional, which is an advantage of this interpretation.

A third interpretation suggests that by saying that the point is one by substance and subject but multiple by being and account, Rufus may be making a distinction like that between extension and intension.61 The infi nity of points in a longer line does not exceed the infi nity of points in a shorter line extensionally, but it might be considered greater intensionally, in virtue of the way the infi nities are conceived or re-presented, by using them as endpoints of line segments or measuring them in some other way.

Against this suggestion there are considerations related to medieval and modern philosophical usage: Rufus has at hand medieval technical

60 I owe much of this interpretation to class discussion and particularly to Joshua Snyder. He and Shawn Burns wrote a paper based on this interpretation of Rufus, “Rufus on the Comparison of Infi nities,” which may subsequently appear in print. The example of line segments comes from Josh. A similar suggestion was made by Giorgio Pini: given a shorter (AC) and a longer line (AD), point C at the end of the shorter line will be one in account, since it serves only as the end of AC. But the point cor-responding to C in AD will be two in account in AD, since it can be counted both as the end of AC and the beginning of CD. Thus there will be at least one more point in account in AD than in AC.

61 I owe this suggestion to Gary Ebbs and much of the discussion to Timothy O’Connor and Krista Lawlor.

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terminology to describe such distinctions and does not use it here. On a variety of topics, Rufus distinguishes between signifi cation and supposition or appellation, where signifi cation corresponds roughly to sense, and appellation to extension or reference. Similarly, Rufus speaks of formal predication when he wants to distinguish different natures in the same real subject—for example, intellect and will are both identifi ed as the rational soul, but not with each other, so will and intellect differ in formal predication or defi nition. Rufus makes reference to neither distinction here.

Nor does Rufus’s discussion of ways of counting provide the kind of explanation normally provided by the intension-extension distinction today. Moderns often use the distinction between intension and exten-sion when they want to explain changes in reference despite a fi xed intension. Such variation is explained in terms of different concepts. For example, the description ‘animal with heart’ and ‘animal with kidney’ are coextensive—that is they pick out the same creatures; the intensions of the two descriptions differ, however. Or for a medieval Aristotelian, ‘featherless biped’ and ‘rational animal’ are conceptually distinct descriptions with the same supposition or appellation. What Rufus seems to want to explain, however, are not conceptual differ-ences, but differences in the external world. So for example, I think he wants to explain the different ways the point at the center of the world can be numbered if two lines are drawn from it to the circumference rather than three.

In favor of the suggestion is that Rufus seems to think that what varies is how we count things. So, for example, he seems to want to explain how the common sense is one when we only see an object, but two when we both see and touch it. The common sense counts as two if it is receiving sensible species from two senses simultaneously. Unlike many modern usages, it is not our knowing more or less that seems salient here. Rather, the point serves more or fewer functions in counting or measuring.

Will this explain in what sense the point at the center of a circle counts as less than all the points in a maximal line, though infi nitely many lines can meet at any point? A point at the center of a circle is potentially infi nitely many and so are the points in a maximal line. There are in fact more points in a line than at the center of a circle, if only because there is more than one point in any line, more in subject and substance. But, surely, Rufus wants us to understand a sense in which taken together the points in the line are more in account. So perhaps


he is saying that more things will actually be measured by reference to the points in a line than to a point at the center of the circle. Again what will determine how great a point is is not the potentially infi nite number of lines that could meet at it, but the lines we actually mea-sure in reference to it. The “greatness” of a point in this sense will depend on how often it is used in measurement. Whether this should be described as intensional I must leave to others more versed in the modern literature. If we describe as intensional what we use in mea-surement, perhaps it does.

We do not yet, and we may never, have enough evidence to decide be-tween these three interpretations, though I am inclined to think that the second is most likely. In any case, however, the emphasis on the claim that one incomplete infi nity cannot exceed another suggests that the sense of greater-than Rufus wants to admit will involve counting and comparing the completed parts of infi nities, as do each of the suggested interpretations. It looks as if Rufus wants us to compare the parts that correspond to each other in different infi nities to get an acceptable sense in which one infi nity can be greater than another. In the fi rst interpretation, it will take more cuts to reach points at any given distance from each other in a longer than a shorter line. And similarly, points in AD will count as beginning more lines than points in AC, and points that appear in three measurements will be greater than those in two. Rufus’s emphasis on actual cuts or intersections suggests that he is deliberately looking for a fi nite number of diverse accounts for a point and avoiding the incomprehensibility medievals believe would be involved in considering infi nitely many accounts of a point. Hence the example of the common sense that presumably counts as three when it judges that honey is yellow, sweet, and sticky as the terminus of the senses of touch, taste, and sight, but only as two if we are blind-folded. Here having come to a sticky point, we should stop.

4. Conclusion

What have we seen? We have seen that Rufus is sure that points as mathematical entities are in the world, and equally certain that points are not quantitative parts. Both are aspects of agreement with Aristotle but on which Rufus’ exposition shows some progress in the sense of increased clarity. We have also seen that Rufus does not get beyond certain problems in Aristotle; his discussion of intelligible matter seems

64 rega wood

equally obscure. On the other hand, it may have been useful to try to explain how points can be origins or parts of a line. Also an attempt has been made to provide a consistent account of the troublesome defi nition of point provided by the Posterior Analytics.

The infl uence of mathematics, with its indivisibilist assumptions, was as unhelpful to Rufus as to Aristotle.62 But Rufus has been more careful not to commit himself to problematic assertions based on mathematics. Spheres touch only partly at a point; not points but lines fl ow (In Phys.). Lines are composed not of points but of matter arranged pointwise; points are only quasi origins of the line (DMet). These may not be entirely felicitous solutions, but at least they refl ect a consistent aware-ness of the commitments of the philosophical position.

Finally, possibly on account of his Christian commitments, Rufus has a new problem to deal with: how to fi nd an acceptable sense in which to affi rm that though one infi nity does not extend beyond another, nonetheless one infi nity can be greater than another. Whether it had much infl uence, we cannot tell at this point. As John Murdoch has pointed out, however, Olivi may have been aware of a similar posi-tion.63 But no doubt Grosseteste exercised more infl uence than Rufus. Still, Rufus provides an interesting and credible response and a useful starting point for debate. And this is true for his answers to all three of our questions.

62 See Michael White and Wilbur Knorr, as cited by White, “Aristotle on the Non-Supervenience of Local Motion,” pp. 154–155.

63 Murdoch, “The ‘Equality’ of Infi nites in the Middle Ages,” pp. 171–174; Peter of John Olivi, Quaestiones in II Sent. [ Jansen], pp. 30–40.

RICHARD KILVINGTON ON CONTINUITY*

Elżbieta JungRobert Podkoński

Although with his solutions to the problem of the possible existence of indivisibilia Richard Kilvington seems to fi t into the main stream of the fourteenth-century considerations, which leaned toward the refutation of atomism, he attacks and solves the problem in an original manner. In this paper we will focus on two of Kilvington’s questions, respectively from his De generatione et corruptione and Sentences commentaries, where he presents geometric proofs for the infi nite divisibility of a continuum. Richard Kilvington’s commentary on De generatione et corruptione is a set of ten fully developed questions.1 They were written around 1324–1325

* We want to express our gratitude to Chris Schabel for his help with English.1 All ten questions are contained in Mss: Bruges 503, ff. 20vb–50vb (along with

Kilvington’s questions on the Ethics and the Sentences); Erfurt SB Amploniana O–74, ff. 35ra–86va, Sevilla Bibl. Columbina 7.7.13, ff. 9ra–29rb, Paris BNF lat. 6559, ff. 61ra–119va. Part of the set is to be found in Cambridge Peterhouse 195, ff. 60r–69r and Krakow Bibl. Jagiellonska, cod. 648, ff. 40ra–53rb. In Ms. Vat. lat. 4353, the manu-script, where Maier (cf. Maier, Ausgehende Mittelalers, pp. 253–54) ‘found’ Kilvington’s commentary on the Physics, there are 40 lines of the fourth question listed below.

This is a list of questions contained in Ms. Paris BNF lat. 6559: 1. ff. 61ra–65rb: Utrum augmentatio sit motus ad quantitatem (q. 3 in Mss. Krakow BJ.

648 and Bruges 503, Erfurt SB Ampl. O–74). 2. ff. 65va–68va: Utrum numerus elementorum sit aequalis numero qualitatum primarum (q.

7 in Mss. Bruges 503, Erfurt SB Ampl. O–74). 3. ff. 68va–71rb: Utrum ex omnibus duobus elementis possit tertium generari (q. 9 in Mss.

Bruges 503, Erfurt SB Ampl. O–74). 4. ff. 71rb–88rb: Utrum in omni generatione tria principia requirantur (q. 10 in Mss. Bruges

503, Erfurt SB Ampl. O–74; part of the question contained in Vat. lat 4353, f. 125r). 5. ff. 89ra–97vb: Utrum continuum sit divisibile in infi nitum (q. 2 in Mss. Krakow BJ 648,

Bruges 503, Erfurt SB Ampl. O–74). 6. ff. 97vb–101va: Utrum omnis actio sit ratione contrarietatis (q. 5 in Mss. Krakow BJ 648,

Bruges 503, Erfurt SB Ampl. O–74). 7. ff. 101va–105vb: Utrum omnia elementa sint adinvicem transmutabilia (q. 7 in Mss. Bruges

503, Erfurt SB Ampl. O–74). 8. ff. 105vb–112vb: Utrum mixtio sit miscibilium alteratorum unio (q. 6 in Mss. Bruges 503,

Erfurt SB Ampl. O–74). 9. ff. 112vb–119va: Utrum omnia contraria sint activa et passiva adinvicem (q. 4 in Mss.

Krakow BJ 648 and Bruges 503, Erfurt SB Ampl. O–74).10. ff. 131ra–132vb: Utrum generatio sit transmutatio distincta ab alteratione (q. 1 in Mss.

Krakow BJ 648 and Bruges 503, Erfurt SB Ampl. O–74).

66 el bieta jung & robert podko ski

and were well known to his contemporaries.2 Kilvington’s last work, Quaestiones super libros Sententiarum, is a set of eight fully developed questions and four subordinate problems, which are also composed as questions.3 The most probable date for Kilvington’s Sentences lectures is 1332–1334. The problem of the division of a continuum was debated by Kilvington at length in the question Utrum continuum sit divisibile in infi nitum (the fi fth question in our list) and was later recapitulated in the Sentences, in the question Utrum unum infi nitum sit maius alio.4

In Kilvington’s time, the most popular geometrical proofs against atomism were those of John Duns Scotus, which were later adopted by William of Ockham and then developed by Thomas Bradwardine.5 John Duns Scotus presents two anti-atomistic geometrical arguments. He begins the fi rst one with a construction of two concentric circles on

2 The question on reaction (the ninth in our list) inspired Heytesbury and gave him an impulse to debate the problem (on the discussion on the issue see Caroti, “Da Walter Burley al Tractatus sex inconvenientium: la tradizione inglese della discussione medievale De reactione,” pp. 279–331). Maier suggests that Wodeham’s references are a report of Richard FitzRalph’s arguments against Kilvington’s theory of infi nity (on FitzRalph’s polemic with Kilvington, cf. Maier, Die Vorläufer Galileis im 14. Jahrhundert, pp. 208–211; Courtenay, Schools and Scholars in Fourteenth-Century England, pp. 76–78; K. Walsh, A Fourteenth-Century Scholar and Primate: Richard FitzRalph in Oxford, Avignon and Armagh, pp. 19–20).

3 The work, whole or in parts, is contained in the following manuscripts: Bruges 503, ff. 79vb–105rb; Bruges 188, ff. 1–56; Bologna, Archiginnasio A–985, ff. 1a–52a; BAV, Vat. lat. 4353, ff. 1–60; Florence, Bibl. Naz., Magliabecchi II. II 281, ff. 43–50, Paris, BNF, lat. 15561, ff. 198–228; Paris, BNF, lat. 17841, f. 1r–v, Erfurt, CA 2o 105, ff. 134–81; Prague, Univ., III. B. 10, ff. 191–227; Tortosa, Cat. 186, ff. 35r–66r. For detailed information on secondary literature cf. Kretzmann, The Sophismata of Richard Kilvington, p. XXVI, n. 35.

The titles of the questions are as follows:1. Utrum Deus sit super omnia diligendus;2. Utrum per omnia meritoria augeatur habitus caritatis quo Deus est super omnia diligendus; a. Utrum aliquis possit augmentare peccatum alteri;3. Utrum omnis creatura sit suae naturae cum certis limitatibus circumscripta; a. Utrum aliquod corpus possit simul et semel esse in diversis locis; b. Utrum unum infi nitum sit maius alio;4. Utrum quilibet actus voluntatis per se malus sit per se aliquid;5. Utrum peccans solum per instans mereatur puniri per infi nita instantia interpellata; a. Utrum voluntas eliciens actum voluntatis pro aliquo instanti debeat ipsum actum per aliquod

tempus necessario tenere;6. Utrum aliquis nisi forte in poenam peccati possit esse perplexus in hiis quae pertinent ad

salutem;7. Utrum omne factum secundum conscientiam ab aliquo sitt meritorium;8. Utrum peccatum veniale aggravet mortale mortaliter.4 For detailed information about Kilvington’s works and secondary literature see

Jung-Palczewska, “Works by Richard Kilvington,” pp. 184–225.5 Murdoch, “Thomas Bradwardine: Mathematics and Continuity in the Fourteenth

Century,” pp. 104–110; Podkoński, “Al-Ghazali’s Metaphysics as a Source of Anti-Atomistic Proofs in John Duns Scotus Sentences Commentary,” pp. 614–618.

richard kilvington on continuity 67

the basis of the third postulate of book I of Euclid’s Elements.6 Then, assuming that the circumferences of the circles are composed of points, Scotus indicates two points of the circumference of the greater circle that are immediately adjacent to one another. Next, he draws a line from each of those points to the centre of the circles, again invoking the appropriate postulate from the Elements.7 Further, he posits a ques-tion whether these lines intersect the circumference of the smaller circle at one or at two points. If one accepts the latter answer, one must conclude that there are as many points on the circumference of the smaller circle as on the circumference of the greater one, which is obviously absurd. When one agrees, however, that the supposed two radii intersect the circumference of the smaller circle at the same point, then let us draw—Scotus argues—a tangent to the smaller circle from this very point. One of Euclid’s postulates assures us that the tangent is perpendicular to each of the radii.8 Consequently, we obtain two right angles that are unequal, which is also absurd (fi g. 1).

William of Ockham, who generally neglected rationes mathematicae in his philosophical inquiry, brings up a simplifi ed version of Scotus’s argu-ment in one of his Quaestiones Quodlibetales entitled Utrum linea componatur ex punctis (Whether a line is composed of points).9 If we draw all of

6 John Duns Scotus, Ordinatio, [ Balic], Liber Secundus, dist. II, pars II, quaestio 5, pp. 278–350: Utrum angelus possit moveri de loco ad locum motu continuo. Actually, Scotus referred here to the second postulate and it might have been numbered so in the copy of Elements he had at hand, but in modern editions of Euclid the postulate he invoked is the third one (cf. Ibid., note T3). Cf. also Euclid, Elements [trans. Heath], vol. 1, p. 199.

7 John Duns Scotus, Utrum angelus. . . ., op. cit., p. 292: “Super centrum quodlibet, quantumlibet occupando spatium contingit circulum designare, secundum illam peti-tionem 2 I Euclidis. Super igitur centrum aliquod datum, quod dicatur a, describantur duo circuli: minor, qui dicatur d,—et maior b. Si circumferentia maioris componitur ex punctis, duo puncta sibi immediata signentur, quae sint b c,—et ducatur linea recta ab a ad b et linea recta ab a ad c, secundum illam petitionem I Euclidis ‘a puncto in punctum lineam rectam ducere’ etc. Istae rectae lineae, sic ductae, transibunt recte per circumferentiam minoris circuli. Quaero ergo aut secabunt eam in eodem puncto, aut in alio?”

8 John Duns Scotus, ibid., pp. 292–293: “Si autem duae rectae lineae ab et ac secent minorem circumferentiam in eodem puncto (sit ille d), super lineam ab erigatur linea recta secans eam in puncto d, quae sit de,—quae sit etiam contingens respectu minoris circuli, ex 17 III Euclidis. Ista, ex 13 I Euclidis, cum linea ab constituit duos angulos rectos vel aequales duobus rectis,—ex eadem etiam 13, cum linea ac (quae ponitur recta) constituet de angulos duos rectos vel aequales duobus rectis; igitur angulus ade et etiam angulus bde valent duos rectos,—pari ratione angulus ade et angulus cde, valent duos rectos. Sed quicumque duo anguli recti sunt aequales quibuscumque duobus rectis, ex 3 petitione I Euclidis; igitur dempto communi (scilicet ade), residua erunt aequalia: igitur angulus bde erit aequalis angulo cde, et ita pars erit aequalis toti!”

9 William of Ockham, Quodlibeta Septem [ Wey], Quodlibet I, Quaestio 9, pp. 50–65, Utrum linea componatur ex punctis.


the radii from the centre of any two concentric circles to each of the indivisibles that constitute the circumference of the outer circle—says Ockham—these radii should intersect both circles in the same number of constituent indivisible points. Therefore any two such circles must be equal in circumference, which is obviously false (fi g. 2).10

10 William of Ockham, ibid., pp. 54–55; Cf. also: Podkoński, “Al-Ghazali’s Metaphysics as a Source of Anti-Atomistic Proofs in John Duns Scotus Sentences Commentary,” pp. 614–615.

Fig. 1

an indivisibleline

a tangent to the smaller circle

e

D

B

a

d

rightangles

b c

Fig. 2

Indivisible lines


The second of Scotus’s anti-atomistic geometrical arguments concerns the incommensurability of the diagonal and the side of a square. Generally, the construction employed in the following part of the proof is similar to the one presented in Roger Bacon’s Opus Maius, written in 1266 or 1267. We fi nd here the following reasoning:

[ If ] the world is composed of an infi nite number of material particles called atoms, as Democritus and Leucippus maintained . . . the diagonal of the square . . . and its side would be commensurable . . . For if the side has ten atoms, or twelve or more, then let the same number of lines be drawn from those atoms to the same number in the opposite side, the sides of the square being equal; . . . therefore since the diagonal passes through those lines, and no more can be drawn in the square, the diagonal must receive a single atom from each line, and thus they have an aliquot part as a common measure, and the side has just as many parts as the diagonal, both of which conclusions are impossible.11 (fi g. 3).

In the beginning of his proof the Subtle Doctor invokes again the postulates from the Elements that defi ne the notion of geometrical com-mensurability.12 First, he considers the case when parallel lines intersect

11 Roger Bacon, Opus Maius [trans. Burke], p. 173.12 John Duns Scotus, Ordinatio, [ Balic], Liber Secundus, dist. II, pars II, quaestio

5, pp. 297–298.

Fig. 3

Indivisible lines


the diagonal of a square at every point of its length. Next, he posits a hypothesis that there are points on the diagonal that do not belong to any of the parallel lines. If we accept this—Scotus argues—then let us draw a line from any of these points that is parallel to the nearest of the parallel lines assumed before. This new line necessarily crosses the side between the points that constitute it. Consequently, one can indicate a point that lies between two immediately adjoining points. This way one arrives at a contradiction, therefore one must deny the hypothesis that any continuum is composed of immediately conjoined indivisible entities (fi g. 4).13 William of Ockham, in the above-mentioned question, limits himself to a short presentation of Bacon’s argument:

[ If a line were composed of points]—Ockham argues—then a side of a square would be equal to its diagonal and the diagonal would be com-mensurable with the side. The consequence is obvious, because one can draw a line from any point of one side [of a square] up to any point of the opposite side . . . and each of these lines contains a certain point of the diagonal. Consequently, there exists a line between any point of the

13 John Duns Scotus, ibid., p. 298: “Secunda probatio est ex 5 sive ex 9 X Euclidis. Dicit enim illa 5 quod ‘omnium quantitatum commensurabilium proportio est ad invicem sicut alicuius numeri ad aliquem numerum’, et per consequens—sicut vult 9—‘si lineae aliquae sint commensurabiles, quadrata illarum se habebunt ad invicem sicut aliquis numerus quadratus ad aliquem numerum quadratum’; quadratum autem diametri non se habet ad quadratum costae sicut numerus aliquis quadratus ad aliquem numerum quadratum; igitur nec linea illa, quae erat diametri quadrati, commensurabilis erit costae illius quadrati. Minor huius patet ex paenultima I, quia quadratum diametri est duplum ad quadratum costae, pro eo quod est aequale quadratis duarum costarum; nullus autem numerus quadratus est duplus ad alium numerum quadratum, sicut patet discurrendo per omnes quadratos, ex quibuscumque radicibus in se ductis. Ex hoc patet ista conclusio, quod diameter est assymeter costae, id est incommensurabilis. Si autem lineae istae componerentur ex punctis, non essent incommensurabiles (se haberent enim puncta unius ad puncta alterius in aliqua proportione numerali); nec solum sequeretur quod essent commensurabiles lineae, sed etiam quod essent aequales,—quod est plane contra sensum. Accipiantur duo puncta immediata in costa, et alia duo opposita in alia costa,—et ab istis et ab illis ducantur duae lineae rectae, aequidistantes ipsi basi. Istae secabunt diametrum. Quaero ergo aut in punctis immediatis, aut mediatis. Si in immediatis, ergo non plura [sunt] puncta in diametro quam in costa; ergo non est diameter maior costa. Si in punctis mediatis, accipio punctum medium inter illa duo puncta mediata diametri (illud cadit extra utramque lineam, ex datis). Ab illo puncto duco aequidistantem utrique lineae (ex 31 I); ista aequidistans ducatur in continuum et directum (ex secunda parte primae petitionis I): secabit costam, et in neutro puncto eius dato, sed inter utrumque (alioquin concurreret cum alia, cum qua ponitur aequi-distans,—quod est contra defi nitionem aequidistantis, quae est ultima defi nitio posita in I). Igitur inter illa duo puncta, quae ponebantur immediata in costa, est punctus medius.”


diagonal and any point of the side. Therefore, if there were six points composing the diagonal, there would necessarily be six [points] in each of the sides.14

Kilvington, however, does not mention any of these constructions in his works. Among twelve principal arguments from his question Utrum continuum sit divisibile in infi nitum, which appeal to the problem of infi nite divisibility from mathematical, physical, and metaphysical points of view, one fi nds three strictly geometrical proofs. In these arguments Kilvington deals with the following examples: an angle of tangency (angulus contingentiae), “a cone of shadow” and a spiral line. For this reason, in order to expose his theory of continuity we will present his detailed deliberations on these three study cases. Since Kilvington goes back to his geometrical proofs only once in his later work (in his questions on the Sentences), we will also attempt to answer the question about the consistency of his theory.

14 William of Ockham, Quodlibeta Septem [ Wey], Quodlibet I, Quaestio 9, p. 51: “Si sic, tunc costa esset aequalis dyametro et esset diameter commensurabilis costae. Consequentia patet quia a quolibet puncto dyametri ad costam contingit protrahere lineam rectam. Quod patet quia a quolibet puncto costae unius ad quodlibet punc-tum costae alterius contingit protrahere lineam rectam, immo ita esset de facto posita hypothesi; et quaelibet talis linea protrahitur per aliquod punctum dyametri; igitur a quolibet puncto dyametri ad costam est aliqua linea recta. Si igitur sint sex puncta in dyametro, erunt necessario sex in utraque costa.”

Fig. 4

Immediate points in a side of a square

a “mediate” point in a side of a square

a “mediate” point in a

diagonal of a square


1. An Angle of Tangency

Kilvington begins all his analyses of the concept of continuity with the claim that every continuum, especially a mathematical one, is infi nitely divisible. In the eleventh principal argument he claims that one of the geometrical “entities” introduced by Euclid himself does not fi t this Aristotelian concept of continuity. The case considered is an angle of tangency. In fact, there is only one defi nition within Euclid’s work that is devoted to this “species” of angles, i.e. proposition 16, Book III. It reads:

The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be inter-posed; further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilinear angle.15

Consequently Euclid, as it seems, introduces an infi nitely small math-ematical being that seems to be an indivisible.

While referring to the above defi nition, Kilvington does not cite it verbatim, but gives his own interpretation, saying that “an angle of tangency is not divisible by a straight line but by a circular one.”16 First, Kilvington tries to convince the reader that Euclid is wrong, and in order to do so he sets up the following geometrical construction. Let’s take two contiguous circles, one of which is two times greater than the other, and a line that is contiguous to both of them in the point of contact. When the line is rotated around the point it will fi rst cross this bigger circle without crossing the smaller one. This will be so because—argues Kilvington—all the points of the bigger circumference are closer to the line than the points of the smaller one.17 Therefore

15 Euclid, Elements [trans. Heath], vol. 2, p. 37.16 Richard Kilvington, Utrum continuum sit divisibile in infi nitum, Erfurt SB Ampl. O–74,

f. 41vb: “Si quaestio foret vera, igitur angulus contingentiae foret divisibilis in infi nitum. Consequens falsum et contra conclusionem 14 III Euclidis, ubi dicitur quod angulus contingentiae non est divisibilis secundum lineam rectam sed circularem.”

17 Richard Kilvington, ibid., f. 41vb: “Probo, quod sit divisibilis secundum lineam rectam, quia capio duos circulos quorum unus sit duplus ad alium et contingant se. Et capiatur linea contingens illos circulos in eodem puncto in quo circuli se contingant, et pono quod illa linea quiescat secundum illum punctum et moveatur secundum aliud extremum. Quo posito, arguo sic: si illa linea secet utrumque circulorum praedictorum, et propinquior est maiori circulo secundum omnia sua puncta quam minori, igitur citius secabit maiorem circulum quam minorem. Quo concesso arguo sic: haec linea secat circulum maiorem et non minorem, igitur illa dividit angulum contingentiae contentum a circulo minori.”


we will come up with a rectilinear angle that seems to be smaller than an angle of tangency (fi g. 5).

Kilvington’s own response to this argument reveals that he is abso-lutely aware that the conclusion is false. He simply says that the line will cross both circles, the bigger and the smaller, simultaneously.18 Although Kilvington does not offer any explanation, it seems that he knows that every straight line that crosses the point of tangency and forms an angle with a tangent is a secant (fi g. 6.). Consequently, no rectilinear acute angle is smaller than an angle of tangency. This is, most likely, why he concludes that an angle of tangency can be divided only by circular lines.19

Next, Kilvington observes that both an angle of tangency and a rectilinear angle are infi nitely divisible. Evidently, Kilvington’s statement stems from the following reasoning: since both angles are infi nitely divis-ible, one can recognize them as infi nite sets of infi nitely small parts that constitute the angles; and in accordance with Aristotle’s opinion it was commonly accepted that all infi nities are equal, so in this sense an angle

18 Richard Kilvington, ibid., f. 43rb–va: “Et dico, quod acceptis duabus lineis cir-cularibus, quarum maior contineat intrinsecus minorem, et capiatur linea in puncto contactus et moveatur secundum aliud extremum, dico quod non prius secabit illa linea circulum maiorem quam minorem.”

19 Richard Kilvington, ibid.: “Ad undecimum principale, quod angulus [contingentiae] est divisibilis secundum lineas circulares et non rectas.”

Fig. 5

α


of tangency and a right angle are equal; therefore, one would say that there is a proportion between them, namely an equal proportion.20

But in his answer Kilvington states that there is no proportion between any rectilinear angle and an angle of tangency, because: “this angle is an infi nite part of a right angle, like a point is an infi nite part of a [fi nite] line”.21 Again, Kilvington leaves the readers with the diffi culty of reconstructing the possible reasoning that underlies his conclusion. The above statement is in accordance with Campanus of Novara’s claim that “any rectilinear angle is greater than an infi nite number of angles of contingency.”22 And in this case, as it seems, Kilvington denies the existence of any proportion between the multitudes of rectilinear acute angles and of curvilinear angles in order to avoid any method of constructing the acute angle that would be equal to the angle of tangency. If we accepted that there are as many “parts” of the right angle as “parts” of an angle of tangency, we might recognize some kind of correspondence between rectilinear acute angles and curvilinear ones. And consequently, the “most acute” of these angles would equal the “smallest” of the curvilinear angles—only when it is agreed that the “smallest” angles exist. But Kilvington is of the opinion that any

20 Richard Kilvington, ibid., f. 41vb: “Item, si sic, cum angulus rectus sit divisibilis in infi nitum secundum lineas rectas proportionaliter, igitur aliquis angulus rectus haberet proportionem ad angulum contingentiae, et ita angulus rectus et angulus contingentiae forent aequales.”

21 Richard Kilvington, ibid.: “Item, si angulus contingentiae foret divisibilis secun-dum lineam circularem, igitur angulus contingentiae haberet aliquam proportionem ad angulum rectum. Consequens falsum, quia iste angulus est infi nita pars anguli recti, sicut punctus infi nita pars lineae.”

22 John Campanus of Novara, [in:] Murdoch, “The Medieval Language of Proportions: Elements of the Interaction with Greek Foundations and the Development of New Mathematical Techniques,” p. 243.

Fig. 6

α


continuum is infi nitely divisible, and consequently there is no last and “smallest” part of it.23

Kilvington accepts Ockham’s defi nition of a continuum, according to which every continuum is regarded as a set containing in actu infi nite subsets of smaller and smaller proportional parts.24 In the case consid-ered a right angle is such a continuum that is divisible into smaller and smaller acute rectilinear angles. At the same time, each of them is bigger than an angle of tangency, and an angle of tangency is also infi nitely divisible into proportional parts. The fact that both angles should be considered as an infi nite set and subset allows us to state that one is bigger or lesser than the other, but it does not allow us to establish any proportionality between them.25 This claim is also in accordance with Ockham’s concept of relations between actual infi nities that must be unequal.26 Evidently, Kilvington recognizes an angle of tangency as an actually infi nitely small geometrical “entity,” which can be divided, however, in infi nitum.27

23 Richard Kilvington, Utrum continuum . . ., Ms. Paris, BNF lat. 6559, f. 95ra: “Ad rationem, quando quaeritur et cetera, dico, quod continuum est duplex sicut quan-tum, quia aliquod est continuum per se et aliquod per accidens. Continuum per se est illud, quod per se est quantum et illud est tale, quod habet divisiones quae sunt accidentia sua. Alio modo sumitur continuum per accidens, et sicut dicimus quod albedo est continuum. Divisio etiam accipitur dupla. Uno modo pro illo, cuius partes possunt actualiter dividi sive separari per divisionem. Alio modo pro eo, quod habet partes quae possunt separari ab invicem sive per divisionem sive non, et sicut dicimus quod coelum est continuum et non divisibile quia partes eius non possunt ab invicem separari. Sed primo modo accipiendo continuum et hoc modo continuum qualiter-cumque intelligendo quaestionem universaliter quaestio est falsa, accipiendo secundo modo quaestio est vera.”

24 For a detailed explanation of Ockham’s conception of infi nity, see Goddu, The Physics of William of Ockham, pp. 159–176. Kilvington’s considerations on infi nity are presented in Podkoński, “Thomas Bradwardine’s Critique of Falsigraphus’s Concept of Actual Infi nity,” pp. 147–153.

25 Richard Kilvington, Utrum continuum . . ., Erfurt, SB Ampl. O–74, f. 41vb: “Item, si sic, cum angulus rectus sit divisibilis in infi nitum secundum lineas rectas proportion-aliter, igitur aliquis angulus rectus haberet proportionem ad angulum contingentiae, et ita angulus rectus et angulus contingentiae forent aequales. Consequentia probatur. Sequitur, quod nullus angulus sit maior angulo recto, et per consequens sequitur, quod angulus rectus sit infi nitus, cum habeat infi nitas partes proportionales quarum nulla pars unius est pars alterius, et quaelibet est maior angulo contingentiae.”

26 William of Ockham, Comm. Sent. II, q. 8, as quoted in Murdoch, “William of Ockham and the Logic of Infi nity and Continuity,” p. 170: “concedo quod infi nita essent excessa, sicut probat ratio, et quod unum infi nitum esset maius alio.”

27 Ricardus Kilvington, Utrum continuum . . ., Erfurt, SB Ampl. O–74, ff. 42rb–43va: “Ad undecimum principale, quod angulus est divisibilis secundum lineas circulares et non rectas. Et nego consequentiam: igitur angulus contingentiae habet certam propor-tionem cum angulo acuto vel recto. Et dico, quod acceptis duabus lineis circularibus,


2. “A Cone of Shadow”

At the beginning of the ninth principal argument Kilvington says that according to Plato there is a kind of fi gure that is indivisible.28 As a matter of fact, in his Timaeus Plato claims that all beings are made up of solids and solids are made up of indivisible triangles that form their sides.29 First, Kilvington states that there is a triangle whose apex is contingent to its base.30 In order to prove his assumption, Kilvington presents the following mental experiment. Let us imagine that a lucid body A illuminates an opaque, circular, fl at body B that is smaller than A. In effect, we obtain a cone of shadow—C. Then, let’s presume that while A increases in size, external parts of B are continuously becom-ing transparent. The process of transmutation of B occurs according to its proportional parts, i.e in the fi rst period of time one half of all radii of B diminishes, and in the next period of time one half of the remaining parts of radii diminishes, and so on (fi g. 7).31

Then Kilvington simplifi es the case taking into account only one of the vertical sections of the cone C: a triangle formed by one of the diameters of the remaining part of B and two rays of light tangent to it. Thus the height of this triangle is continuously decreasing until B is wholly transparent. Now, it is easier to consider the case as a series of smaller and smaller triangles (fi g. 8). Eventually, Kilvington observes that,

quarum maior contineat intrinsecus minorem, si accipiatur linea contingens utrumque circuli et quiescat in puncto contactus et moveatur secundum aliud extremum. Dico quod non prius secabit illa linea circulum maiorem quam minorem.”

28 Richard Kilvington, ibid., f. 41ra: “Aliqua superfi cies est indivisibilis sicut Plato ponit.”

29 See Plato, Timaeus, 53c. 30 Richard Kilvington, Utrum continuum . . ., Erfurt, SB Ampl. O–74, f. 41ra: “aliquis

est triangulus cuius unus punctus est immediatus eius basi, et omnia puncta eius sunt immediata.”

31 Richard Kilvington, ibid., f. 41 vb: “Igitur radii incidentes a partibus circumferen-tialibus ipsius a per partes circumferentiales ipsius b concurrunt. Consequentia patet, quia illae lineae non sunt aeque distantes, et cum illae non concurrant in b nec citra b, igitur concurrunt ultra b. Capio igitur piramidem ex partibus illarum linearum quae sunt ultra, qua piramidis sit c. Tunc c est obumbrata quia nullae lineae incidunt ab a ad aliquod punctum intrinsecum ipsius c. Tunc sic: si b continue corrumpetur in corpus dyaphanum, et prius secundum partes circumferentiales quam centrales, et a continue quiescat non auctum nec transmutatum, igitur partibiliter illuminatur. Consequentia patet, quia lineae incidentes ab a per partes b erunt infra lineas incidentes ab a per puncta extrinseca b, igitur erunt breviores illis. Et secabunt se in aliquo puncto c, et cum in quolibet instanti post initium in quo incipit b corrumpi erunt lineae incidentes ab a per partes b; igitur in quolibet instanti postquam b inceperit transmutare erit plus illuminatum de c.”


if the above-described process terminates, one of these triangles must be the last and as such it is the smallest and indivisible. It is obvious that the smallest triangle consists of three points—the apex and two extremes of the basis—that are immediately adjacent to one another.32 It is clear that this assumption is based on the concept of discontinuity stemming from Aristotle’s Physics, that any two immediately adjacent points are contingent and do not constitute a unity.33

In the next paragraph, Kilvington asks whether it is possible for cone of shadow C to disappear totally when there remains some parts of B.

32 Richard Kilvington, ibid., f. 41rb: “Capio tunc aliquod dyametrum partis non corruptae de b, tunc dyameter cum partibus linearum incidentium per extrema illius dyametri causant unum triangulum, qui non est divisibilis, quia tunc esset aliqua pars medii ultra b non illuminata, quod est contra positum.”

33 Cf. Aristotle, Physics, VI, 1, 231a–b.

Fig. 7

Fig. 8

B

r

r/2

r/4

B

C

A


He considers the case in a manner that is typical of his other works, establishing a kind of adequacy between proportional parts of a certain period of time, namely one hour, and succeeding stages of transmuta-tion of A and of B. Again, he simplifi es the case examining separately the growth of A, B remaining unchanged, and the diminishing of B, when A does not change its dimensions.34

In his answer Kilvington states that the fi nal result is the same in both instances: there must always be a cone of shadow left. Consequently, when A and B are changing simultaneously, as long as B exists there is a cone of shadow, no matter how small it is.35 Eventually, Kilvington concludes that the whole case is founded on a false assumption, because no surface is indivisible. He does not present, however, any explanation referring to his—above-described—mental experiment. Most likely he does so because it is obvious for him and his audience that, although there will fi nally be a cone of shadow formed by immediately conjoined points, it does not mean that its section could be recognized as an

34 Richard Kilvington, Utrum continuum . . ., Erfurt, SB Ampl. O–74, f. 41rb: “Et probo, quod totum c illuminabitur antequam b corrumpatur, quia b corrupto et a non aucto in eodem <tempore> corrumpetur b et illuminabitur c. Igitur a aucto et corrupto b, prius illuminabitur c quam b corrumpatur. Antecedens de se patet, et probatur consequentia, quia ex corruptione b tantum illuminabitur de c in uno tempore sicut in alio sibi aequali, et ex augmentatione similiter a, igitur tantum est illuminatum de c per augmentationem a et corruptionem b in uno tempore sicut in tempore sibi aequali. Ponatur igitur, quod b corrumpatur in hora. Tunc sic: in medietatae horae erit plus quam medietas c illuminata per augmentationem a, igitur corrumpitur b et uniformiter illuminabitur c post medietatem horae sicut in prima parte. Igitur in minori tempore quam sit medietas horae illuminabitur pars residua de c. Consequentia patet. Et antecedens probo, quia si non, tunc maioretur. Et tunc in medietatae horae esset medietas c illuminata, et sic plus erit illuminata propter maiorationem a, igitur in eodem tempore erit plus quam medietas illuminata. Antecedens patet, quia si a non augmentetur tunc illuminabitur totum c in hora et c uniformiter illuminabitur; igitur in medietate temporis erit medietas illuminata, igitur erit illuminata propter maiorationem a. Patet, quia si in principio corrumpatur medietas c illuminata sive tanta pars per quantam corruptionem illuminabitur c et deinde maioretur a, tunc per maiorationem a erit aliquid illuminatum de c et tantum erit illuminatum. Si augeretur quando b corrumpetur, igitur et cetera.”

35 Richard Kilvington, ibid., f. 43rb: “Et dico, quod isto casu posito, quod non prius terminabitur c quam corrumpatur b. Et nego istam consequentiam: a non aucto et b corrupto, c illuminabitur in hora, igitur a aucto et b corrupto illuminabitur c ante fi nem horae. Et causa est quia augmentatio ipsius a non facit c citius illuminari secundum se totum, quam illuminabitur post corruptionem b. Sed quod ante fi nem horae plus illuminetur per a, si augeretur, et tamen non sequitur, quod c citius illuminabitur per augmentationem a et corruptionem b, quam per corruptionem b tantum.”


extended surface. In the other part of this question Kilvington states explicitly that two immediate lines do not enclose a surface between them.36

3. A Spiral Line

The third purely geometrical case, Kilvington debates, is based on the following statement: each infi nite line, as continuous, is infi nitely divis-ible. This claim stands against the commonly accepted opinion that an infi nite line cannot exist, because, if it did, each of its parts would be infi nite and thus a part would equal the whole. First, Kilvington pres-ents a procedure for creating an infi nite line. Let’s take a column—says Kilvington—and let’s mark out all its proportional parts: a sequence of halves of its height. And then we draw a spiral line, winding around this column, starting from a point on a circumference of its base, in such a way that each succeeding coil embraces one proportional part, i.e. the fi rst coil the fi rst half of a column, the second one fourth, the third one eighth, and so on in infi nitum (fi g. 9). It is obvious that each coil is longer than a circumference of the column, and there are infi -nitely many coils forming one line. This line is evidently continuous because an ending point of one coil is the beginning of the next coil. Consequently, the line is actually infi nitely long, for it can be regarded as a sum of infi nitely many parts, each of them possessing a certain longitude.37

36 Richard Kilvington, ibid., f. 38ra–b: “aliqua est linea quae ducitur a puncto assignato ad duo puncta immediata, vel non sed duae. Non secundo modo quia forent immediatae. Sequitur quod duae lineae immediatae claudunt aliquam superfi -ciem—quod est impossibile.”

37 Richard Kilvington, ibid., f. 39rb–va: “Si quaestio est vera, tunc linea infi nita foret divisibilis in infi nitum. Consequentia patet, quia continuum est divisibile in infi nitum et linea est continua, igitur et cetera. Minor patet, scilicet quod aliqua linea sit infi nita, quia sit aliquod corpus columpnare a, tunc in a est aliqua linea infi nita. Quod probo sic: quia capio aliquam lineam gyrativam ductam super primam partem proportionalem ipsius a, quae sit b; b ergo est quantitas continua cuius convenit addere maiorem partem in infi nitum quarum nulla est pars alterius nec econverso, igitur convenit devenire ad aliquam lineam infi nitam actu. Quod probatur: consequentia prima per Aristotelem III Physicorum et Commentatorem, commento 64, ubi probant additionem in continuo per partes aequales in infi nitum, et arguant sic: si talis additio sit possibilis, convenit devenire ad aliquam aliam magnitudinem infi nitam in actu. Sic arguo in propositio. Et primum antecedens probo, quia convenit addere ipsi b lineam girativam secun-dae partis proportionalis, tertiae et quartae et sic in infi nitum, quarum quaelibet est


Next Kilvington notes that it is impossible to construct an infi nite line in the described way, because a column circumvolved by a spiral line is of a fi nite height. Consequently the spiral line must have two extremes, i.e. its beginning and end points, and as such it must be fi nite. Thus, Kilvington attacks the problem of labeling an end point of a spiral line. In the fi rst step he takes, he proves that a spiral line has to be immediately adjacent to the upper surface of the column, because if it were not, there would be some proportional parts of the column not circumvolved by the spiral line. Consequently, the spiral line would be fi nite, because it would consist only of a fi nite number of coils. But if a spiral line is immediately adjacent to the upper surface, it should be possible to label its end point. And if there were an end point the line would be fi nite, which is against the main proposition of this argument.38

continua alteri, igitur convenit sibi addere lineas infi nitas aequales, quarum quaelibet est maior c, et quarum nulla pars unius est pars alterius. Consequentia patet, quia cuiuslibet partis proportionalis linea girativa est maior linea circulari secundum quam attenduntur girationes corporis.”

38 Richard Kilvington, ibid., f. 39va: “Item, hoc probo per rationem quod impossibile

Fig. 9

CE

B

A

D

h/

2 h/

4 h/

8


Kilvington proves that although there is “no distance” between the spiral line and the upper surface of the column, there is also no determined point ending the line. In fact, he observes, one can consider any of the points on the circumference of the upper surface of the column as an end point of the spiral line. And if it is so, it has an infi nite number of end points.39 Therefore, there is no end point, and the line is infi nite and immediate to the circumference of the upper surface of the column.

In his answer Kilvington affi rms the last conclusion and says that the spiral line has two limits. One of them is intrinsic—and this is the starting point of the line. The other, however, is an extrinsic limit—and this is the circumference of the upper surface of the column, which does not belong to this line. The only possible explanation is that Kilvington considers the spiral line to approach this circular line asymptotically (fi g. 10).

Kilvington repeats the above-presented construction of an infi nite spiral line in his question Utrum unum inifnitum potest esse maius alio from his commentary on Peter Lombard’s Sentences. It serves as one of many obvious examples of actual infi nities that are to be found or are pos-sible in the created world. The difference is that Kilvington shows a method of constructing the spiral line that is infi nite with respect to both of its extremes. He draws two spiral lines, both starting from the same point in the middle of the height of a column and going into

est aliquam lineam esse infi nitam. Quia pono tunc quod talis foret in corpore colump-nari, et sit illa b. Capio igitur lineam circularem in extremo a corporis circularis, versus quod sit progressio partium proportionalium—qua sit c, tunc si b sit linea infi nita est immediata c extremo. Quia si b et c distarent aliqua esset proportionalis pars inter b et c—et sic hoc solum componeretur ex partibus fi nitarum partium proportionalium et sic b foret fi nita—quod est contra positum.”

39 Richard Kilvington, ibid., f. 39va–b: “Sed probo quod b et c non sunt immediata, quia b terminabitur ad aliquod punctum ipsius c et non est maior ratio quare magis ad unum quam ad aliud; igitur b terminatur ad infi nita puncta. Huic dicitur quod b terminatur ad punctum tantum, qui terminat lineam rectam a quam incepit. Verbi gratia, posito quod prima giratio incepisset in d puncto de lineae, tunc necessario ter-minabitur ad punctum e terminum eiusdem lineae. Sed contra, probo per rationem, quod terminabitur ad quodlibet punctum, quia giratio primae partis proportionalis secat omnem lineam rectam in superfi cie illius corporis extrema protensam ab uno extremo in aliud. Capio igitur aliquam linearum—et sit illa linea a, igitur et ponatur quod secet eam in hoc puncto fa lineae. Tunc in hoc puncto incipit una giratio quae terminabitur ad eandem lineam et punctus girans non recedet a fa linea nisi iterum accedat ad ipsam. Igitur, si ille punctus omnino girat a corpus per modum puncti, sequitur quod in fi ne sit d punctus in f puncto fa, igitur b linea terminabitur ad f punctum, et per idem potest probari quod terminaretur ad quolibet punctum.”


opposite directions toward its upper surface and base.40 The laconic manner of presenting this construction suggests that Kilvington pre-sumes that this argumentation is familiar to his audience. As it was already proven in the question Utrum continuum, both halves of the line would lack an end point, and consequently the whole line would be infi nite utroque extremo.

4. Conclusions

It is clear at the outset, that in his question on the continuum Kilvington does not take part in a discussion on indivisibilism. Although Kilvington, like John Duns Scotus, William of Ockham and Thomas Bradwardine, employs geometry, he is instead only interested in revealing paradoxes resulting from the Aristotelian defi nition of continuity. If we take for granted that a continuum is infi nitely divided into proportional parts, we have to accept that there is no last proportional part. Consequently, the process of dividing cannot be completed and lacks its limit.

40 Ricardus Kilvington, Utrum unum infi nitum potest esse maius alio, Ms. BAV, Vat. lat. 4353, f. 40r–v: “linea sit infi nita utroque extremo, ut patet de linea gyrativa in corpore columpnari quae per utriusque suae medietatae girat singulas partes proportionales versus extrema illius corporis. Et quod talis sit infi nita patet, quia additio fuit sibi per partes aequales in infi nitum, igitur ibi est infi nitum tale et in actu.”

Fig. 10

2π 4π 6π 8π 10πx

h


All three of the above geometrical examples show that the Archimedean principle of continuity is not valid here. This principle, repeated later on by Euclid, states that:

Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continu-ally, there will be left some magnitude which will be less than the lesser magnitude set out.41

In the fi rst case discussed, Kilvington shows that there is no transi-tion between an angle of tangency and a rectilinear one, although both are certain geometrical magnitudes. The second case—“a cone of shadow”—demonstrates that if a process of the diminution of a magnitude is continuous, it cannot end. It seems that in the third case discussed one fi nds the results of the fi rst two, since it employs both a proportional division and a transition between two different kinds of geometrical entities—a spiral and circular lines. It also shows that we cannot come to the end, because we cannot fi nd the last point of a spiral line.

These examples are modifi cations of Zeno’s paradox, namely the paradox of dichotomy. Even though Aristotle was certain that while introducing isomorphism of different kinds of continua he eliminated the paradox,42 Kilvington makes it clear that Aristotle failed. As a matter of fact, the Aristotelian defi nition of continuity seems to be in accordance with Zeno’s statement that infi nite division into propor-tional parts cannot be completed. But Kilvington is not able to solve this paradox. He only notices contradictions that derive from accepted principles, and he leaves the readers—as it is apparent—with the diffi culty of interpretating his mathematical arguments. Kilvington, however, does not accept the alternative solution according to which the process of the division of a continuum can be completed because there are indivisible entities. But he explicitly argues for the commonly accepted Aristotelian concept of continuity.

Only the last of Kilvington’s above-presented geometrical construc-tions exercised the interest of other medieval thinkers. One fi nds the debates on properties of a linea girativa in the works of e.g. Roger Roseth,

41 Euclid, Elements [trans. Heath], vol. 3, X, prop. I, p. 258.42 Cf. Aristotle, Physics, VIII, 10, 266b2.


John Buridan, Benedict Hesse and John Major. Most of them, however, took into account only one aspect of Kilvington’s discussion, asking whether a spiral line is actually or potentially infi nite.43

43 Cf. Roger Roseth, Utrum aliqua creatura possit esse infi nita, in Lectura super Sententias, Quaestiones 3, 4 & 5 [ Hallamaa], pp. 266–272; John Buridan, Utrum linea aliqua gyrativa sit infi nita, et semper accipio infi nitum categorematice, in: Quaestiones super libros Physicorum, Liber III, Quaestio 16, edited in Thijssen, John Buridan’s ‘Tractatus de infi nito,’ pp. 23–33; Benedict Hesse, Utrum aliqua linea gyrativa sit infi nita, accipiendo ‘infi nitum’ categorematice, in Quaestiones super octo libros Physicorum Aristotelis [ Wielgus], pp. 384–387; John Major, De infi nito [ Élie], pp. 12–52.

THE IMPORTANCE OF ATOMISM IN THE PHILOSOPHY OF GERARD OF ODO (O.F.M.)*

Sander W. de Boer

Introduction

The Franciscan theologian Gerard of Odo (Giraldus Odonis) was bornc. 1285 in the village of Camboulit, near Figeac in the South of France, and died in 1349.1 He lectured on the Sentences in the Franciscan studium in Paris in the period 1326–1328.2 Odo wrote an infl uential commentary on Aristotle’s Nicomachean Ethics (to which he owes his name of Doctor moralis).3 Even though he did not write more commentaries on the works of Aristotle, Odo did have a great interest in logical and natural-philo-sophical topics. There are several separate questions or tracts in these fi elds, most of them anonymous, that have been ascribed to him. Most of this material can also be found in some form in his commentary on the Sentences.4 Since Odo has not written any commentary on the

* I would like to thank Prof. Paul J.J.M. Bakker and Prof. Hans M.M.H. Thijssen for their helpful comments on an earlier version of this article.

1 For biographical details see: Schabel, “The Sentences Commentary of Gerardus Odonis, O.F.M.”; De Rijk, Giraldus Odonis O.F.M.: Opera Philosophica I: Logica, pp. 1–5 and Weijers, Le travail intellectuel à la Faculté des arts de Paris: textes et maîtres (ca. 1200–1500), III, pp. 79–83.

2 These Parisian lectures were actually the second time Odo lectured on the Sentences. The fi rst time was in Toulouse in the late 1310s. Parts of these Toulouse lectures must have found their way into the Parisian commentary, but it is unknown how much. For a detailed description of Odo’s commentary on the Sentences, cf. Schabel, “The Sentences Commentary of Gerardus.”

3 This infl uence can, for example, clearly be traced in John Buridan’s commen-tary on the Ethica as James Walsh has demonstrated in Walsh, “Some Relationships between Gerald Odo’s and John Buridan’s Commentaries on Aristotle’s ‘Ethics’.” The commentary survives in about 17 Mss. For a list of the Mss and early prints of this commentary see Lohr, ‘Medieval Latin Aristotle Commentaries. Authors G–I’, p. 164. The Venice edition (1500) is accessible on Gallica (http://gallica.bnf.fr/).

4 Cf. De Rijk, “Works by Gerald Ot (Gerardus Odonis) on Logic, Metaphysics and Natural Philosophy Rediscovered in Madrid, Bibl. Nac. 4229.” For examples of this connection between the tracts in the Madrid Ms and the Sentences commentary, cf. Bakker, “Guiral Ot et le mouvement. Autour de la question De motu conservée dans le manuscrit Madrid, Biblioteca Nacional, 4229” and De Rijk, Giraldus Odonis O.F.M.: Opera Philosophica II: De intentionibus.

86 sander w. de boer

Physics, these separate tracts combined with his commentary on the Sentences are the only sources we have to determine his views on topics of natural philosophy.

In this article I want to examine his position on the question in natural philosophy that he is most famous for, both in his own time and ours, namely the question of the structure of the continua: space, time and motion. Understanding the structure of continua was so important to Odo, that he dedicated two extensive questions to it in his commentary on the Sentences. In addition, there are two separate tracts on this issue that are closely related to these questions in the Sentences. All his treat-ments of the problem have a very similar structure and the core of his solution is always the same: continua are composed of a fi nite number of non-extended indivisibles that touch each other whole to whole. The reason why indivisibles can compose something that is extended in this way, is that they have, as Odo calls it, certain differences of either place or time.5 When indivisibles touch each other whole to whole, but not according to every difference, they can compose something extended. Besides this notion of ‘difference’, Odo also introduces the principle that there can be local motion without the moving thing changing position. This occurs, for example, when a sphere is rotated around its centre, according to Odo. In such a situation only the differences of the point in the centre would “move”. With this solution, Odo is the fi rst of the Parisian atomists in the fourteenth century.6

After the pioneering research on fourteenth-century atomism by Pierre Duhem and Anneliese Maier, there have been two scholars in particular who have signifi cantly contributed to our knowledge of Odo’s atomism.7 The fi rst was Vassili Zubov who in 1959 published an article

5 “. . . quod indivisibile secundum partes quantitativas est distinguibile et determinabile secundum differentias respectivas loci vel temporis.” (Ms Madrid, Bibl. Nac., 4229,f. 183vb)

6 This group includes: Nicholas Bonetus, Marc Trivisano, John Gedeonis, Nicholas of Autrecourt. Nicholas Bonetus includes many parts of Odo’s text almost verbatim in the part on quantity in his Liber predicamentorum (Venice, 1505), even if he eventually reaches a somewhat different version of atomism; cf. Zubov, “Walter Catton, Gérard d’Odon et Nicolas Bonet.” Another follower is Marc Trivisano, who includes large parts of Odo’s text in book 2 of his tract ‘De macrocosmo’; cf. Boas, “A Fourteenth-century Cosmology.” The otherwise unknown John Gedeonis, to conclude, as John Murdoch pointed out, also keeps referring to the arguments of a certain magister (undoubtedly Odo) when he develops his atomistic position in Ms Vat. Lat. 3092, ff. 113v–124r.

7 Duhem and Maier could only consult Odo’s tract on the continuum that is con-tained in Ms Vat. Lat. 3066. The Ms contains only the recto side of the fi rst folium of a question on the structure of the continuum. In order to determine Odo’s position,

atomism in the philosophy of odo 87

on the atomism of Gerard of Odo, Walter Chatton, and Nicholas Bonetus.8 Zubov was mainly interested in the atomist replies to a series of mathematical arguments against the possibility of atomism. To give an example of one such argument against atomism: given a square and its diagonal, we can draw a line from each point on the side of the square, through the diagonal, to its corresponding point on the opposite side. The next step in the argument is to ask what would happen if we were to draw all the lines from all the points on the side. Within an atomistic conception of magnitude, there are only two options. If each such line intersected precisely one point of the diagonal, the side and the diagonal would not only be commensurable but would be exactly the same in size. If, on the other hand, there were points on the diagonal that had not been intersected after drawing all the pos-sible lines, a line could be drawn through these non-intersected points on the diagonal from one side of the square to the other. However, in that case, the assumption that all possible lines have been drawn, is false. Thus, a magnitude cannot consist of atoms.

These mathematical arguments gained their fame through Duns Scotus’s commentary on Peter Lombard’s Sentences.9 All these arguments, which derived from the Arabic tradition and entered the Latin tradition through Al-Ghazali’s summary of Avicenna’s critique of atomism, point to the incompatibility of atomism with the truths of Euclidean geom-etry.10 All late medieval atomists felt they had to respond to them.

The second scholar who has made a major contribution to our understanding of Odo is John Murdoch, in studies devoted to late-medieval atomism.11 Even though the focus of his research has been

they therefore had to rely mainly on refutations of Odo’s views, in particular by John the Canon in book VI of his commentary on the Physics. Cf. Maier, Die Vorläufer Galileis im 14. Jahrhundert, pp. 162–163, and Duhem, Le système du monde: histoire des doctrines cosmologiques de Platon à Copernic, VII: La physique parisienne au XIV e siècle, pp. 403–412.

8 V.P. Zubov, “Walter Catton, Gérard d’Odon et Nicolas Bonet.” 9 Scotus, Ordinatio [ Balic e.a.], dist. 2, pars 2, q. 5: “Utrum Angelus possit moveri de

loco ad locum motu continuo.” The corresponding question in the Reportata Parisiensa gives the arguments in an abbreviated form. The arguments are not given in the cor-responding question in the Lectura.

10 The arguments are given in the fi rst part of the capitulum de diversitate senciendi de composicione corporis; Algazel, Metaphysica [ Muckle], pp. 10–13.

11 Important publications (the list is not exhaustive) are: Murdoch, Geometry and the Continuum in the Fourteenth Century: A Philosophical Analysis of Thomas Bradwardine’s Tractatus de Continuo; idem, “Superposition, Congruence and Continuity in the Middle Ages;” idem, “Naissance et développement de l’atomisme au bas Moyen Âge latin”; idem, “Infi nity and Continuity”; idem, “Atomism and Motion in the Fourteenth Century;” Murdoch


on the atomists in Oxford, he has written about Odo as well. Murdoch has classifi ed fourteenth-century atomism as a doctrine that is not the result of a physical analysis, but rather of a purely intellectual reaction to Aristotle’s analysis of continuous quantity.12 This would not only explain the fact that the atoms were thought of as similar to math-ematical points, but also the fact that even the relations between these atoms were conceived in mathematical terms. This view also leads to the conclusion that fourteenth-century atomism is not a description or explanation of reality, as was the case in Greek atomism.

Because Murdoch focused on those parts of Odo’s tract that illus-trated the late-medieval application of propositional and mathematical analysis the other parts were not discussed. In particular, Murdoch stud-ied the responses of Odo and other atomists to the (often mathematical) arguments against their atomistic position. As a result, Odo’s positive arguments in favour of atomism have never been examined in detail. It is precisely on this point that I hope to contribute to our knowledge of his atomism. Starting my analysis with Odo’s arguments in favour of atomism, I intend to complement the picture of Odo’s atomism that we had so far, and also bring out its ontological ramifi cations.

In this article I will focus on the separate tract on the continuum that is found (only) in the Ms Madrid, Biblioteca Nacional, 4229, to which I will refer in the rest of this article as De continuo.13 The tract is closely related to Odo’s treatment of the continuum in book II of his commentary on the Sentences.14

1. My Theses

In this paper, I hope to show two things about the position and impor-tance of atomism in Odo’s philosophy, namely, fi rst, that his atomism occupies a much more important and central place in his philosophy

& Synan, “Two Questions on the Continuum: Walter Chatton (?), O.F.M. and Adam Wodeham, O.F.M.”

12 Murdoch, “Naissance et développement de l’atomisme,” p. 27.13 I have also looked at Odo’s other texts on the continuum. Although there are

some interesting differences between the different versions, they have no implications for the conclusions of this article.

14 The precise relations between the different versions of Odo’s questions on the structure of continua are complex. I’m currently preparing an article on this topic which will also include a critical edition of the De continuo tract from the Ms Madrid, Bibl. Nac. 4229.


than we assumed, and, secondly, that this atomism is not just a purely intellectual response to Aristotle’s treatment of continuous quantity, but also makes an ontological claim. More precisely, in my view Odo intended his atomism to be both an accurate description and an expla-nation of real continuous physical processes and structures.

To substantiate these claims, I also want to propose a new interpre-tation of his atomism in which the two following principles play an important role. (1) In Odo’s atomism there is an ontological priority of the part over the whole, i.e. the whole is seen as nothing more than the sum of its parts, and every property of the whole can be reduced to the properties of the parts. (2) The necessity of a fi nite number of indivisibles follows from the ontological primacy of the part and therefore is not motivated by the necessity of a limit to division, but by the composition of continua out of prior parts combined with the (Aristotelian) idea that the infi nite cannot be traversed.

To make clear what I mean by this second principle, let me make a distinction between what I would call a strong and a weak reading of the term `composed’ in the proposition ‘a continuum is composed of indivisibles’. In the weak reading this proposition only means that a continuum can be divided into indivisibles. Therefore, a continuum is in some way composed of them since these indivisibles had to be somehow (at least potentially) contained in the continuum prior to the division. This weak reading always takes the whole existing con-tinuum as its starting point, and therefore never implies a primacy of the indivisible parts over the whole. It is in exactly this same way that the non-atomist will claim that a continuum is composed of parts that are always further divisible, without implying in any way that these parts exist prior to the whole continuum.15 That is, if we divide an existing continuum, we can always divide the resulting parts further ad infi nitum. The strong reading, on the other hand, does imply a primacy of the part. In this reading a continuum is composed, that is formed, by a continuous joining of prior existing part to part. It is this strong reading that describes the position of Odo. To avoid any confusion on what this primacy entails, we could say the following. The ontological primacy of the part, however crucial in understanding Odo’s atomism, does not imply any form of Democritian atomism where all the atoms fi rst exist separately. It does imply that the answer to the question “how

15 Cf. Aristotle, Physics VI, 232b24–26.


do the indivisible parts of the continuum come into being?” is not “by division of the continuum,” but is “they come into being, one by one, when the continuum is composed”.

It is important to note that even though an atomism that is an expla-nation of the physical world, such as the atomism of Democritus or Epicurus, needs some sort of primacy of the indivisible part, an atom-ism that is only the result of an intellectual analysis of the structure of continuous quantity does not. Such an “analytical” atomism could be motivated solely by the need for a limit to divisibility.

2. The ‘De continuo’ Tract

Let me now turn to Odo’s De continuo tract. In this text Odo gives six arguments in favour of his atomism as well as six arguments against it. Here I will discuss two of his arguments in favour of atomism, the fi rst and the sixth.

2.1. The Distinctions between Act-Potency and Quantitative-Proportional

The fi rst argument in De continuo is also the most important one, because it introduces a number of distinctions that play a major and recur-rent role in Odo’s refutation of a number of counter-arguments. The argument also shows the primacy of the part in Odo’s atomism. The basic argument is very brief and runs as follows: every whole composed of an infi nite number of magnitudes, just as a cubit is composed of two semi cubits, is an actual infi nitely large magnitude. And since an actual infi nite magnitude is impossible, no continuum can be infi nitely divisible.16

The phrasing of the argument is important here. Where the fi rst premise speaks of “being composed” (compositus), the conclusion speaks of “being divisible” (divisibilis). This implies, if the argument is valid, a parallelism between composition and divisibility, in the sense that something is divisible in the same number of parts as it is composed of. Given this parallelism the continuum indeed cannot be infi nitely

16 De continuo, ff. 179rb–va: “Omne totum compositum ex magnitudinibus multitudine infinitis, sicut componitur cubitus ex duobus semicubitis, est magnitudo actu infinita. Sed non est dare magnitudinem actu infinitam. Ergo nullum continuum est divisibile in infinitum.”


divisible, since an infi nite number of composing parts would make the continuum infi nitely large.17 The addition of the clause “as a cubit is composed of two semi cubits” is crucial for this line of reasoning, as we will see.

In its basic form the argument is far too crude, as Odo himself also realizes. He therefore introduces two objections, both of which argue that some distinction needs to be made. The fi rst objection states that being composed of parts can be understood in two ways. In the fi rst way we understand the parts to be actually present, and in the second way we understand the parts to be merely potentially present. The infi nite number of parts in a continuum must be understood in the second, potential, way.18

Odo’s analysis of this distinction between act and potency is best seen in the passage where he distinguishes between two meanings of a potentiality of parts.

. . . this division by act and potency either distinguishes between the manner in which the infi nite number of parts exist in the continuum, in the sense that the infi nite number of parts do not actually exist in the continuum but only potentially, or distinguishes between the manner of multiplication of the parts of the continuum, in the sense that all the parts of the continuum taken together are not to be actually multiplied in infi nity, but in potency only, because they are not actually multiple given the fact that they are not actually divided.19

I will start with the second meaning, which Odo calls “the multiplica-tion of parts”. In this case we take all the parts together, that is we start from the unity of the continuum, and say that there is no actual infi nity of parts. The core of Odo’s defence is that this second read-ing no more denies an actual infi nity of parts than it denies an actual

17 There is one hidden premise, which is that there is a smallest part in this infi nity. In the way Odo thinks about composition, a composition from a collection of parts where there is no smallest one is ruled out, as will be seen.

18 De continuo, f. 179va: “Ad hanc rationem dicentur duo secundum duas distinctiones communes. Primo dicendo quod componi aliquid ex magnitudinibus multitudine infi nitis potest intelligi dupliciter. Primo actu, secundo potentia. Nunc autem ita est quod con-tinuum est compositum ex magnitudinibus multitudine infi nitis in potentia, non tamen in actu. Et ideo non oportet continuum esse magnitudinem actu infi nitam.”

19 De continuo, f. 179va: “. . . haec divisio per actum et potentiam vel distinguit inesse partium infinitarum, ita quod sit sensus quod partes infinitae non insunt actu continuo sed potentia tantum, vel distinguit multiplicationem partium continui, ita quod sit sensus quod omnes partes continui simul sumptae non sunt actu multiplicandae in infinitum sed in potentia, quia non sunt actu multae exquo non sunt actu divisae.”


duality or triplicity.20 In other words, we take the unity of the continuum and claim that there is no actual multitude. Odo simply dismisses this reading as not responding to his own argument, since this reading does not speak of parts but only of unity. It is important to note that the reason why he can so easily dismiss the objection taken in this sense is precisely his phrasing of his own argument in such a way that it starts from the priority of the parts. This priority is secured by the addition of the clause “as a cubit is composed of two semi cubits”.21

The second reading of the difference between actual and potential in the argument, does start from a primacy of the part. In this read-ing the potentiality concerns the manner in which the infi nity of parts is present in the whole. Odo’s response to this reading is surprisingly short. He immediately claims that in this reading the objection is false, because an infi nite number of parts would make the whole infi nitely large. Although he gives no explanation here, we can reconstruct the underlying idea. For what could the potency of the part mean, if a continuum is composed out of prior existing parts? It could mean nothing more than the parts being continuously joined, and therefore not actually being distinguishable as parts, since they do not exist as actually separate from the whole continuum. Interpreted in this way, the difference between act and potency of the part cannot solve any-thing, since an infi nite number of parts joined together constitutes an infi nite whole, whether the original composing parts are distinguishable in the whole or not.

To summarize, the validity of Odo’s whole argumentation against the distinction between act and potency is based on the assumption that continua are composed of ontologically prior parts. For if the two semi cubits were not prior but only arose from the division of the cubit, Odo could no longer reject the counter-argument that the infi nity of parts in the continuum is merely potential. Recall that the only reason why he could so easily reject this reading of the distinction between act and potency, was that it started from the unity of the continuum and therefore did not respond to his argument.

20 De continuo, f. 179vb: “Non ergo plus vitatur per illam solutionem infi nitas partium quam dualitas vel quaternitas. Quare illa solutio non valet.”

21 Of course the distinction I made between a weak and a strong reading of “being composed” could also be applied to this clause. It is, however, clear that Odo takes the clause in the strong reading where we fi rst have two semi cubits and only after combining those a cubit. Otherwise his argument would make no sense.


His imaginary opponents, however, have another way out. This is the Aristotelian distinction between two types of division, namely between quantitative and proportional division.22 Where in a quantitative divi-sion a whole is divided into a certain, always fi nite, number of equally sized parts, in a proportional division, a whole is divided according to a certain factor, usually the factor two, in a potentially infi nite number of parts decreasing ever in magnitude, but without a last and minimal part. This distinction gives Odo’s opponents a powerful objection, since it concerns the manner of division of the continuum and therefore always starts from the unity of the continuum.

Surprisingly, in his response to the objection, Odo denies that there is a difference between these two ways of dividing. To understand why he denies this, we must look at the peculiar way in which he understands the proportional division. For Odo, a proportional division means that I take the whole and divide it into two parts. Then I take both parts and divide them both into two parts. Then I take the resulting four parts and again divide each into two parts, and so on.23 Now it is evident that the result of this procedure (at any stage of the division) is a fi nite number of equal parts. And taken this way, a proportional division is little more than an unnecessarily complicated way of describing a quantitative division. Also, since a proportional division now amounts to the same as a quantitative division, an infi nite proportional division would result in an infi nite number of parts of equal magnitude. And this would imply, as Odo says, that the whole continuum would be infi nite in size.24 The emphasis on the resulting equal magnitude of the parts is meant to exclude the possibility of a proportional division in the

22 De continuo, f. 179vb: “Alio modo dicetur ad rationem dicendo quod aliquid com-poni ex magnitudinibus multitudine infi nitis contingit intelligi dupliciter. Primo quod infi nitae magnitudines illae sint eiusdem quantitatis. Vel secundo: eiusdem proportionis. Et secundum hoc dicitur quod compositum ex magnitudinibus multitudine infi nitis, si sint eiusdem quantitatis, est infi nitum actu. Si vero non, sed eiusdem proportionis, non erit magnitudo infi nita. Nunc autem ita est de continuo quod componitur ex magnitudinibus infi nitis eiusdem proportionis, non eiusdem quantitatis. Ideo non est magnitudo infi nita actu.”

23 De continuo, f. 179vb: “. . . quia infi nitae magnitudines eiusdem proportionis neces-sario sunt eiusdem quantitatis. Quod patet, quia accipiatur una quantitas et dividatur in duas quantitates aequales; iterum illae dividantur in alias equales et sic in infi ni-tum, semper multiplicatio secundum illam proportionem dividetur secundum eandem quantitatem inter se.”

24 De continuo, f. 179vb: “Omne compositum ex magnitudinibus secundum quantita-tem aequalibus et secundum multitudinem infinitis, sicut cubitus componitur ex duobus semicubitis, est magnitudo actu infinita.”


Aristotelian sense. Of course, if my interpretation of Odo’s atomism is correct, such proportional division has to be excluded on the grounds that it has no last part, and therefore the parts could never be prior to the whole but would always be posterior.

In conclusion, we have seen that Odo’s fi rst argument is explicitly formulated as an argument that starts from the composition of the continuum out of prior parts of equal magnitude. Without the addition of the phrase: ‘as a cubit is composed of two semi cubits’ his argument would fail. In his response to the objections, all distinctions are reduced to a resulting fi nite number of parts of an equal magnitude.

2.2. Degrees of Heat

To show that this priority of the parts is fundamental for the whole of Odo’s atomism and not just implied in his fi rst argument, I want to discuss a second argument he gives in favour of his atomism. It is the sixth and last argument in De continuo. It discusses what occurs in the intension of heat.25 Here Odo argues that given an infi nite divisibility of the continuum, heat would also be infi nitely divisible; and therefore there would be an infi nite intensity of heat. For heat is caused by a qualitative motion, and this motion, being continuous, would include an infi nite number of “having changeds” (mutata esse). In each of these mutata esse, the intensity of the heat would increase, no matter how little. In a similar manner as in the fi rst argument, Odo concludes that this infi nite number of increases of intensity would result in the fi nal intensity of the heat being infi nitely great.26

The phrasing of the argument is again far from innocent. The premise stating that in each of the mutata esse the intensity of heat will increase implies fi rst of all that there can be motion in a single moment, something Aristotle explicitly denies;27 and secondly it also seems to

25 De continuo, ff. 182ra–b: “Sexta ratio principalis est haec: si continuum esset divisibile in infi nitum, calor esset divisibilis in infi nitum et esset actualiter infi nitus. Utrumque consequens est impossibile. Ergo et antecedens. Quod primum consequens sit falsum: dicit Philosophus De sensu et sensato quod nulla passio sensibilis est in infi nitum divisibilis. Aliud consequens est manifeste impossibile.”

26 De continuo, f. 182rb: “Arguo sic: cuicumque calori additi sunt infiniti calores faci-entes unum cum ipso, est actualiter calor infinitus. Sed si motus calefactionis <esset> divisibilis in infinitum habens infinita mutata esse per quae sunt acquisiti infiniti gradus caloris, necesse est quod primo calori sint additi infiniti gradus et per consequens infiniti calores. Ergo quilibet calor intensus per motum esset actualiter infinitus.”

27 Aristotle, Physics [ Barnes], VI, 241a15–16: “Again, since motion is always in time and never in a now, and all time is divisible (. . .).”


imply an even stronger claim, namely that a change or movement in a certain time-interval is only possible when there is motion or change in each of the moments in that interval; and therefore that the properties of a continuum originate from the properties of its indivisible parts. In other words, if there were no change in each indivisible part of the motion, there could be no change in the whole motion. The parallel with the fi rst argument for atomism will be clear. Again, Odo tries to show that an infi nite number of parts in a continuum will always constitute an infi nite whole.

And just as in the fi rst argument Odo lets his opponents object with the distinction between actual and potential parts. The primacy of the parts is perhaps most clearly stated in the following passage:

I argue against this: I accept that all the degrees that are acquired by movement are actually acquired and actually exist in the heat. I accept from the other part <of the syllogism> that these degrees of form are actual. Then <I argue> as follows: an infi nite number of actualities that are actually acquired render that in which they are actually infi nite. But as is clear from what was presupposed, these degrees are actual, are actually acquired and are infi nite in number. Therefore they render something actually infi nite, because this absence of distinction does not remove the infi nity. In fact, if there were an infi nite number of distinct cubits, <and> if afterwards they were joined and became indistinct, such an absence of distinction would not at all prevent that this infi nite number of cubits would render the quantity actually infi nite.28

In this fragment Odo responds to the objection that we need to distin-guish between act and potency, and that the parts in the intension of heat are only potential parts. According to Odo, this distinction can-not solve anything, since the infi nite number of degrees are at some moment actually received. And from that moment on they are all contained in the unity of the resulting heat. That the resulting heat is a unity, in which the composing degrees are no longer distinguishable does not detract anything from the fact that, even as undistinguished, they all contribute to the intensity of the resulting unity. And to leave

28 De continuo, ff. 182rb–va: “Contra istud arguo: accipio quod omnes gradus acquisiti per motum sunt actu acquisiti et actu insunt calori. Accipio ex alia parte quod isti gradus formae sunt actuales. Tunc sic: infinitae actualitates numero actu acquisitae reddunt istud cui insunt actu infinitum. Sed ut patet per supposita, isti gradus sunt actuales et actu acquisiti et secundum multitudinem infiniti. Ergo ipsi reddunt actu aliquid infinitum, quia illa indistinctio non tollit infinitatem. Si enim essent infi niti cubiti distincti, si postea coniungerentur et essent indistincti, talis indistinctio minime prohiberet quin illi cubiti infi niti redderent quantitatem infi nitam actu.”


no doubt, Odo once again brings his example of a cubit into play. If I were to combine an infi nite number of cubits into a longer length, then, even supposing that the individual cubits are not distinguishable in the resulting length, this resulting length will still be infi nite.

Again we see the ontological priority of the part appear in this argument. The importance of this primacy will become clear in the remainder of this article where I intend to show the important role of atomism in Odo’s philosophy and theology. Still, we might ask at this stage why this peculiar way of looking at the relations between parts and wholes in a continuum has not been noted earlier in the scholarly literature? The answer seems to be that it only becomes apparent in the arguments Odo gives in favour of atomism. And those were the arguments that had for understandable reasons not yet received any attention. As for the examined parts of Odo’s atomism, that is his responses to the various counter-arguments, these parts are, as far as I can see, indifferent to the question whether the part or the whole of a continuum is primary.

What is also remarkable about this second argument is that it seems to describe a concrete physical process, which is something we would not immediately expect if Odo’s atomism were indeed primarily moti-vated by a conceptual analysis of continuous quantity. However, it is yet too soon to draw any conclusions here about possible ontological commitments of Odo’s atomism, or its explanatory powers, since the argument occurs in a tract that deals with continua in general and could therefore just as well be a merely hypothetical example.

2.3. A Missing Argument: God’s Omnipotence

There is one fi nal aspect of De continuo I want to draw attention to in this context, namely the lack of arguments invoking God’s omnipo-tence, that were both common and important in the fourteenth-century atomism debate, for example in the writings of the Oxford atomists Walter Chatton and Henry of Harclay.29 In De continuo there are no such

29 For Chatton, cf. Quaestio de continuo [Murdoch & Synan], § 36, pp. 58–59, 61, 68, 72. Also, William of Alnwick attributes two arguments for atomism to Henry of Harclay, in his Determinatio II, both of which invoke God’s omnipotence. As far as I know the Determinatio II is unpublished, but J. Murdoch gives a partial translation (including these arguments) based on his own private edition in Grant (ed.), A Source Book in Medieval Science, pp. 319–324.


arguments.30 Omnipotence arguments were powerful tools for atom-ists to make their atomism plausible. For example, from the argument that God can perceive any point on a given line, it could be concluded that the line was composed of points and therefore had an atomistic structure. Why does Odo not use such powerful arguments?

The common structure of the arguments that start from God’s omnipotence is that even if we cannot effectuate a complete division of the continuum or enumerate every indivisible in it, God can. Therefore a possible (partial) explanation of the lack of such arguments in Odo would be the following. If Odo is indeed convinced of the primacy of the part in continua, as I have argued, then the arguments starting from God’s omnipotence become much less relevant. If the part is always primary, there is no need for a supernatural power to arrive at all the indivisible parts of the continuum, since these indivisible parts are already there, each contributing to the whole. The parts are only potential in the sense that they are not distinguished from each other and do not exist apart from the whole continuum.

Of course, this explanation about the lack of arguments deriving from God’s omnipotence can never be more than a hypothesis. However, that Odo has little need to introduce them to arrive at the indivisible parts of a continuum is clear.

3. Other Contexts

Now that we have a global picture of the manner in which Odo argues for his atomism, and especially of the importance of the presupposition of the ontological primacy of the indivisible part, I want to show that this atomism and this way of looking at the relation between parts and wholes is not limited to Odo’s question(s) on the continuum. If it could

30 This applies to Odo’s other texts on the continuum as well. The only minor exception can be found in the question on the continuum in book I of his Sentences commentary, where we can fi nd one small argument invoking God’s omnipotence. The argument only plays a minor role as a (partial) argument to support the larger argument that claims there can be no order between the divisions of the continuum that would limit its divisibility (also found as the 3rd argument in De continuo). “Tertio arguitur ex eadem via sic: omnem realitatem potest Deus separare a quacumque re quae non est illa res, nec dependet ab ea in esse partis. Ergo Deus potest separare quamlibet partem continui ab alia. Et confi rmatur quia: nulla potentia passiva est in rebus naturalibus excedens omnipotentiam Dei. Sed continuum est potentia passiva ad dividi. Ergo non excedit potentia illa potentiam Dei.” (Ms Sarnano, Bibl. Com., E. 98, ff. 103rb–va).


be shown that Odo introduces an atomistic structure of real continua in texts and arguments that do not deal with continua as their main topic, this would indicate that in the particular case of Gerard of Odo we must qualify Murdoch’s general assessment of fourteenth-century atomism.

In the remainder of this article I will examine a total of three con-texts, taken from Odo’s commentary on the Sentences and from another separate tract, in which atomism plays an important and sometimes even crucial role. The fi rst of these contexts consists of the questions that deal with infi nity.

3.1. Infi nity

Odo discusses infi nity several times in his commentary on the Sentences. His most detailed exposition is found in a question on the possibility of eternal motion.31 Odo’s answer to this question is affi rmative, to the extent that motion could have existed from eternity. In his response to some of the objections he refi nes both the concept of ‘equal things’ (aequalia) and the concept of ‘infi nite magnitude’ (magnitudo infinita). According to Odo, things can be equal in two ways. First, by being limited at the same points (conterminari ); second, in a negative formula-tion, by not exceeding each other (non excedi). Infi nities can only be equal in the second way of not exceeding each other. The notion of ‘infi nite magnitude’ can even be understood in three senses. The infi nity can result from the absence of a beginning, from the absence of an end, or from the absence of both. Motion can then be infi nite in the sense of the absence of a beginning.32

What this solution shows is that Odo did have some conceptual tools to deal with the paradoxes that result from comparing infi nities. It also

31 Sentences II, dist. 2, q. 2 (Ms Valencia, Cab. 200, ff. 17rb–18ra): “Utrum motus potuerit esse ab aeterno.”

32 Ibid. (Ms Valencia, Cab. 200, f. 18ra): “Modo dico quod nulla magnitudo infi nita per exclusionem ultimi potest esse actu pertransita, quoniam esse actu pertransitum imponit ei terminum, sed non imponit sibi primum. Magnitudo vero infi nita per exclu-sionem primi tantum, non tamen per exclusionem ultimi, potest esse actu pertransita, dum tamen non fuerit totaliter pertranseunda, sicut, si dicetur tempus ab aeterno, praeteritum ponetur infi nitum per exclusionem primi tantum; et ideo non repugnaret sibi esse pertransitum. Futuro autem repugnaret esse pertransitum, si ponetur infi nitum per exclusionem ultimi; eodem modo de magnitudine. Dico igitur quod, si motus fuis-set ab aeterno, magnitudo infi nita per exclusionem primi, non per exclusionem ultimi, fuisse potuisset pertransita.”


shows that he was not troubled by the possible or hypothetical existence of infi nities. This indicates that his atomism does not result solely from his incapacity to deal with the notion of infi nity.

More important for my present purpose, however, is another question on infi nity entitled: “Whether the universe could be infi nitely large”.33 The answer that Odo gives here is negative. In itself this is not surprising since most philosophers at that time would deny it. What is surprising, however, is the way in which Odo responds to the well known argument that God in his omnipotence could create an infi nite magnitude and also both an infi nite multiplicity and intensity.34 For, as the argument runs in Odo’s version of it, all that God can create in a single day he can create in a single moment; and all he creates, he can sustain afterwards. In this way God can create an infi nite multiplicity and an infi nite magnitude since each day he could create a single stone and therefore in an infi nity of days he could create an infi nite number of stones, which he can also join together to create an infi nite magnitude. And if he could create this infi nity in an infi nite number of days, he could also create it in an infi nite number of moments.

There are several ways of dealing with such an argument, for example by pointing out that a physically realized infi nite magnitude is an incoherent notion, since it would occupy a place, and place implies limitation. But Odo takes a very different and unique route as can be seen in the following fragment:

To this I say that just as an act is incompatible with an infi nite magni-tude, so it is incompatible with an infi nite multitude; hence there cannot

33 Sentences II, dist. 44, q. 1 (Ms Valencia, Cab. 200, ff. 97vb–98va): “Utrum machina mundialis possit esse infi nita magnitudine”.

34 Ibid. (Ms Valencia, Cab. 200, ff. 98rb–va): “Sed in oppositum arguitur quia: magnitudo infi nita et multitudo infi nita et virtus infi nita sunt possibiles; ergo mundiali machinae inquantum magnitudo est non repugnat infi nitas; quare ipsa poterit esse infi nita magnitudo. Consequentia est evidens. Sed antecedens probo quia: quaecumque et quantacumque et quocumque Deus potest [potest] producere in diebus infi nitis, <potest> producere in instantibus infi nitis, et illa producta conservare, supposito quod quodlibet illorum sit possibile in instanti. Hoc enim patet, quia quod Deus posset facere uno die, posset facere in instanti, supposita conditione instantanea factionis. Sed Deus posset facere infi nitum magnitudine et multitudine in diebus infi nitis. Quod patet, quia quolibet die posset facere unum lapidem, et per consequens in infi nitis diebus infi nitos lapides, ex quibus resultaret multitudo infi nita. Posset etiam omnia illa copulare continuatione, et esset magnitudo infi nita. Posset etiam quolibet die producere unum calorem, et in infi nitis infi nitos, qui in unum redacti facerent virtutem infi nitam. Quare Deus in instantibus infi nitis posset producere infi nitam multitudinem, infi nitam magnitudinem et infi nitam virtutem.”


be an infi nite number of instants as the minor premiss of the argument supposed. To the argument I say that the minor is false. Neither is the proof valid, because I do not concede that whichever continuum is com-posed of an infi nite number of parts, nor that it is divisible in parts that are always further divisible. On the contrary, I say that it is composed of indivisibles and is resolved in a fi nite number of indivisibles, not an infi nite number.35

Odo denies the existence of an infi nite number of moments! Even if God in each moment creates a stone or a degree of heat and preserves and joins them, the result will always be fi nite, since the number of moments available for this creation is always fi nite. The impossibility of an infi nite multitude is deduced from the non-existence of an infi nite number of moments, that is, from the atomistic structure of time. This means that even from the perspective of God, time has an atomistic structure. And if even God is limited by the atomistic structure of time, interpreting Odo’s form of atomism as solely the result of an analysis of the Aristotelian concept of continuous quantity seems too limited. In this context, Odo uses his atomism as a description of real space and as a partial explanation of the fact this real space cannot be infi nite. That is, his atomism seems to be a physical atomism.

3.2. Intension and Remission

The second context I want to discuss is that of the intension and remission of forms. The relevant questions occur in Odo’s Sentences commentary, but also circulated as a (very long) separate tract, just as the questions on the continuum. Here, I will use the separate tract found in Ms Madrid, Biblioteca Nacional, 4229.

In the fi rst question of this tract, Odo treats the intension of light.36 He proceeds by fi rst stating the opinion of a certain (unnamed) doc-tor, almost certainly Walter Burley, and then giving his own solution.37

35 Sentences II, dist. 44, pars II, q. 1 (Ms Valencia, Cab. 200, f. 98va): “Ad illud dico quod sicut repugnat actus infinitae magnitudini, sic repugnat infinitae multitudini; quare non possunt esse infinita instantia, ut minor rationis supponebat. Ad principale dico quod minor est falsa. Nec probatio valet, quia non concedo quod quodlibet continuum sit compositum ex partibus infinitis, nec sit divisibile in semper divisibilia, ymo dico quod componitur ex indivisibilibus et resolvitur in indivisibilia finita, non infinita.”

36 De continuo (Ms Madrid, Bibl. Nac., 4229, ff. 133ra–135vb): “De augmento formae: utrum lumen augeatur per adventum novae partis ad priorem, utraque manente.”

37 Walter Burley (c. 1275–c. 1346) was a contemporary of Odo and was also (from 1310–1326) connected to the University of Paris. His commentary on the Physics contains an extensive criticism of Odo’s atomism. Also, Burley was one of the most


The opinion of Burley, as described by Odo, is that the intension of light does not occur by the addition of light to light, where both would remain. The argument that is used to disprove such an addition theory is that of a light source continuously moving toward an object in which light is caused. For the continuous movement would mean that in an infi nite number of moments an infi nite number of parts of light is added; and since all the parts would remain, the result would be a light of an infi nite intensity. Just as in the text on the continuum the objection is introduced that the parts exist only potentially. And this objection is again dismissed on the grounds that the resulting intensity is actual and so the composing parts must be actual as well.

Ultimately, Odo accepts Burley’s position, that light is not intensifi ed by an addition of part to part, but not without some critical remarks. Odo says that although Burley’s solution is true and provable, his argu-ment is only ad hominem and not ad rem.38 Now why is the argument only ad hominem? We can infer the reason from the following passage: “Secundam conclusionem, scilicet quod iste doctor demonstraverit conclusionem istam ad hominem concedentem quod continuum sit divisibile in infinitum, probo.”39 The argument is directed against a position that accepts the infi nite divisibility of the continuum, and, so we may add, therefore assumes an infi nite number of moments in a time-interval. The argument cannot be ‘ad rem’ since according to Odo there is only a fi nite number of moments.

After discussing the special case of light, in the next question Odo discusses the intension and remission of qualities. Here Odo’s use of atomism is even clearer. In this question Odo wants to prove, among

prominent defenders of the so-called succession theory in the debate about the intension and remission of forms; a position that Odo will ascribe to the doctor in the rest of his tract. Burley defended this position in a tract called De intensione et remissione formarum [Venice, 1496], which is dated in the 1320’s shortly before Odo’s commentary on the Sentences. Finally, the scribe of the Ms Madrid, Bibl. Nac., 4229 inscribed the name Gal Burley in the margin on f. 142r. For Burley as a defender of the succession theory and for the date of his tract, see Dumont, “Intension and Remission of Forms from Godfrey to Burley” (forthcoming).

38 Tractatus de augmento formae: utrum lumen augeatur per adventum novae partis ad priorem, utraque manente (Ms Madrid, Bibl. Nac. 4229, f. 134va): “Nunc tertio pro evidentia solu-tionis rationum doctoris pono quinque conclusiones. Prima est quod conclusio sua est vera et demonstrabilis ad rem. Secunda quod est demonstrata per eum ad hominem. Tertia quod non est demonstrata per eum ad rem. Quarta quod, dato etiam quod demonstraret, demonstratio sua non esset ad propositum. Quinta quod non respondet ad argumentum factum.”

39 Tractatus de augmento formae . . . (Ms Madrid, Bibl. Nac. 4229, f. 134vb).


other things, that a form is not intensifi ed by addition of degree to degree in an infi nite number of moments. As evidence he gives an example of hot water heating iron.40 According to the addition-theory, at every moment a degree of heat will be caused in the iron. Since the heat in the water is unnatural, the water will continuously cool down, until fi nally it causes a last and minimal (!) degree of heat in the iron and then the heating stops. Now, if there were an infi nite number of moments, and if in each of these moments the water caused a higher degree of heat than the last minimal degree, then according to Odo there would have to be an infi nite number of degrees of heat in the iron when the heating stops. At this point he has already reached his desired conclusion: that there will be an infi nite intensity of heat in the iron and that therefore one of the premises must be wrong.

What is interesting is that the argument continues. In the fi nal step of the argument Odo once again reduces these degrees to equal parts, by saying that an infi nity of degrees, all of which are higher than the last minimal degree, must also imply an infi nity of equal degrees. For the greater, says Odo, is always equal to the smaller plus something extra. Therefore there must be an infi nite number of equal and mini-mal degrees. Note that this fi nal step has only one function, and that is to return to an atomism of minimal parts composing a continuum.

Odo also states the argument in the formal way of the syllogism:

I argue as follows: every form that includes an infi nite number of degrees of the same quantity is infi nite; this <form> is such; therefore it is infi nite. The conclusion is false. Therefore one of the premisses <is false>. Not

40 Tractatus de augmento formae . . . (Ms Madrid Bibl. Nac. 4229, f. 139rb): “Tertiam conclusionem, scilicet quod forma non intendatur per additionem gradus ad gradum per infi nita instantia, probo sic: ponatur aqua calida quae calefaciat ferrum vel aliquam rem neutram, quae non magis sit frigida quam calida; secundum te in quolibet instanti addit gradum alium; et cum aqua habeat calorem innaturaliter et violenter continue minoratur in calore, quia omnis motus violentus fortior est in principio quam in fi ne; minuitur ergo continue in calore, et per consequens in virtute calefaciendi; et ita inducit primo maiorem gradum, et tandem in fi ne non auget amplius calorem; ergo est devenire ad gradum quem ultimo inducit; et iste est minus. Modo si in quolibet instanti inducitur maior gradus, et sint infi nita instantia inter primum instans et ultimum, sequitur quod infi niti erunt gradus maiores isto ultimo, qui est minimus; sed ubi est dare infi nitos gradus maiores, ibi est dare infi nitos gradus aequales, ut patet per Philosophum, quarto Physicorum, quia omne maius est divisibile in partem aequalem et in partem excessus; et sic erit dare infi nitos gradus eiusdem quantitatis praecedentes gradum minimum. Et his premissis, arguo sic: omnis forma includens infi nitos gradus eiusdem quantitatis est infi nita; ista est huiusmodi; ergo est infi nita. Conclusio est falsa.”


the maior premiss. Therefore the minor premiss, that follows from this: that between the fi rst and last <degree> there are an infi nite number of degrees; this follows from another, namely this one: that a form is enlarged by the addition of degree to degree throughout an infi nite number of moments; hence this is false.41

The major premise: every form including an infi nite number of degrees of the same quantity is itself infi nite; the minor: this form (of heat in the iron) is such; and the conclusion: so this form is infi nite. The conclusion is false, so therefore one of the premises, namely the minor. This minor, that the form of heat in the iron has an infi nite intensity, follows itself from the premise that between the fi rst and last degree there is an infi nite number of other degrees. And this in turn follows from the form being intensifi ed by addition of degree to degree in an infi nite number of moments. This last premise then, is false. There is only a fi nite number of moments.

Again Odo’s analysis is consistent with his views on the continuum in De continuo. He consistently analyses concrete physical continuous processes as having a fi nite number of mutata esse, and as occurring in a fi nite number of moments. The topic of the tract in which these passages occur is not continua, but intension and remission. The intro-duction of atomism in this tract must therefore be interpreted as an explanation and description of the structure of physical reality.

3.3. God’s Omnipotence

Perhaps it could be objected at this stage that it might well be the case that Odo uses atomism in contexts that are very closely related to De continuo, like infi nity and intension and remission, but that claim-ing that it is important for the whole of his views, or even the whole of his natural philosophical views, is making too large a claim. To strengthen my conclusions, let me therefore give one fi nal example of how deeply rooted Odo’s atomism actually is. The title of the ques-tion I want to discuss, taken from book I of his Sentences commentary,

41 De continuo (Ms Madrid, Bibl. Nac., 4229, f. 139rb): “. . . arguo sic: omnis forma includens infinitos gradus eiusdem quantitatis est infinita; ista est huiusmodi; ergo est infinita. Conclusio est falsa. Ergo aliqua premissarum. Non maior. Ergo minor, quae sequitur ex isto quod inter primum et ultimum sint infiniti gradus; illa vero ex alia, scilicet ista quod forma intendatur per additionem gradus ad gradum per infinita instantia; quare hoc est falsum.”


is: “on the assumption that God’s omnipotence is infi nite, is it infi nite in respect to an infi nite number of acts or in respect to an infi nitely intensive act?”42

There is no need to discuss the arguments given for both types of infi nity, but it is important to point out that we have already seen all these arguments in the previous texts I discussed. One of the arguments, claiming that possibilities that have no order among themselves can all be actualized at the same time, is even found in the De continuo tract as one of the six positive arguments in favour of atomism.

As we have already encountered these arguments, it is clear what Odo’s answer must be if he is to be consistent. And indeed he denies both forms of infi nity saying the only infi nity that pertains to God’s actions is an infi nity in the sense that no matter how many acts he performs, he can always do more.43 And after explaining this last form of infi nity, Odo concludes with the following passage:

I see what can be answered. For all hold in common that in whichever instant the luminous body is in another place. Secondly, that in whichever instant there is another ray, and it is certain that God can preserve and unite all these rays; hence, because there are an infi nite number of them, one infi nite ray will result. For that reason it seems to me that we must say that in a magnitude there are not an infi nite number of points, nor in motion an infi nite number of having changeds, nor in an hour an infi nite number of moments.44

There is no compelling reason to introduce atomism in a question on God’s infi nity. Odo, however, examines the (hypothetical) results of God using his omnipotence in nature in a way similar to that in which he

42 Sentences I, dist. 44, q. 2 (Ms Sarnano, Bibl. Com., E. 98): “Supposito quod omnipotentia Dei sit infi nita fundamentaliter, utrum sit respectu actuum infi nitorum multitudine vel unius intensive infi niti.”

43 Ibid. (Ms Sarnano, Bibl. Com., E. 98, f. 115ra): “Pro solutione quaestionis, pono tres conclusiones. Quarum prima est quod omnipotentia non potest elicere actus mul-titudine infi nitos. Secundo quod nec actum aliquem extensive vel intensive infi nitum. Tertio quod omnipotentia potest aliquo modo infi nita, quia quotiscumque et quan-tiscumque datis adhoc, potest in plura et in maxima.” The Ms gives ‘infi nite’, which I have corrected to ‘infi nitum’.

44 Ibid. (Ms Sarnano, Bibl. Com., E. 98, f. 115rb): “Video quod possit responderi. Omnes enim communiter tenent quod in quolibet instanti corpus luminosum est in alio situ. Secundo quod in quolibet instanti est alius radius, et certum est quod Deus potest conservare et unire omnes illos radios; quare cum sint infi niti, resultabit unus radius infi nitus. Ideo videtur mihi dicendum quod in magnitudine non sunt infi nita puncta, nec in motu infi nita mutata esse, nec in hora infi nita instantia.”


examines normal natural processes. Nowhere in this question is this approach qualifi ed by a statement like ‘I only speak here as a natural philosopher’ or something of the sort. It seems to me that we can only conclude that Odo was convinced of the real atomistic structure of continua, and that he intended his atomism to be an explanation and accurate description of physical reality.

4. Conclusions

In conclusion, Odo’s atomism turns out to be a constant factor in his philosophy and theology. He consistently uses his atomism to explain reality, and the application of this atomism to God’s power and to the inner structure of continuous physical processes is not provoked by any mathematical argument. On the contrary, mathematics only play a role in countering the common arguments against atomism, as Odo does in De continuo. The main role of Odo’s atomism is to explain what occurs in concrete continuous processes. Given that Odo considers these processes to take place by addition of part to part, where each part contributes to the resulting quality of the whole continuum, it is not surprising he should be convinced that there must be a fi nite number of these basic indivisible parts in each given continuum.

These observations notwithstanding, Murdoch’s general description of atomism in the 14th century as using both mathematical atoms and mathematical relations between them, still is substantially appli-cable to Odo. His atoms are similar to mathematical points, and the relations between them are at least semi mathematical. This results in a tension between a mathematical model and a physical function; a tension that for example becomes clear in the strange and awkward physico-mathematics he uses to counter some of the arguments against his atomism.

The interpretation of Odo’s atomism that I propose also gives rise to the following observation. We know that in Paris another atomist, Nicolas Bonetus, takes substantial parts of his treatment of the con-tinuum from Odo, but sides himself explicitly with Democritus.45 We also know that Nicholas of Autrecourt defends a very physical version

45 Cf. Zubov, “Walter Catton, Gérard d’Odon et Nicolas Bonet,” p. 267.


of atomism.46 From this perspective it would be interesting to see to what extent their positions could be explained as putting more and more emphasis on the physical and ontological side of atomism; a side already clearly present in Odo.

46 Cf. Grellard, Croire et savoir: les principes de la connaissance selon Nicolas d’Autrécourt, in particular the second part.

NICHOLAS OF AUTRECOURT’S ATOMISTIC PHYSICS

Christophe Grellard

In the second chapter of his book Identity and reality, Emile Meyerson, the early twentieth-century French philosopher, provides a picture of what he considered to be some of the perennial characteristics of any atomistic physics.1 His aim is to show that the mechanist conception of natural phenomenona always existed in the history of science. Nicholas’s of Autrecourt fi gured prominently in the list of medieval thinkers who ushered in a new-wave of atomism. Carefully separat-ing Gerard of Odo’s mathematical atomism from Nicholas’s physical atomism, Meyerson’s basic intuition was sound.2 Unlike most medieval atomists, Nicholas did not deploy his atomism in order to solve problems associated with the concept of the continuum.3 His aim was to fi nd a mechanical and reductionist answer to the problem of generation and corruption and, fi nally, to the question of the eternity of things in the world.4 For Nicholas, problems connected with the void and continuum are subordinated to his mechanism. I have claimed elsewhere that Nicholas’s atomism is quite unique, original, especially when compared to other medieval thinkers.5 In this paper, I will demonstrate how atom-ism functions within Nicholas’s physics and how he believes it can resolve some of the basic problems of medieval natural philosophy.

1 Meyerson, Identité et Réalité, p. 90. For a comprehensive study of Meyerson’s con-ception of the history of science, and especially atomism and mecanism, see Fruteau de Laclos, La philosophie de l’intellect d’Emile Meyerson. De l’épistémologie à la psychologie, pp. 42–47 & pp. 62–66.

2 Nevertheless, perhaps should we moderate the mathematical label in the case of Gerard of Odo. See S. de Boer’s contribution in this volume.

3 The mathematical dimension of medieval atomism (seen as a purely intellectual reaction to Aristotle’s conception of continuum) is Murdoch’s main thesis. See Murdoch, “Naissance et développement de l’atomisme au bas Moyen Age latin” and his contribu-tion in this volume. Nevertheless, the cases of Odo, Crathorn, Autrecourt, Wyclif, and even Burley, seem to suggest that we should be more cautious on this point.

4 Cf. Kaluza, “Eternité du monde et incorruptibilité des choses dans l’Exigit ordo de Nicolas d’Autrécourt”.

5 Cf. Grellard, “Les présupposés méthodologiques de l’atomisme: la théorie du continu chez Nicolas d’Autrécourt et Nicolas Bonet”; “Le statut de la causalité chez Nicolas d’Autrécourt.”

108 christophe grellard

Ultricurian atomism begins with the explanation of the eternity of the world. The Exigit ordo’s fi rst chapter deals immediately with this problem and relies on an atomistic conception of nature. Atomism allows Nicholas to criticize Aristotle’s defi nition of generation and cor-ruption as the movement from being to non-being. Nicholas’s atomism is primarily physical and only secondarily mathematical. In the chapter dedicated to the continuum (called De indivisibilibus by O’Donnell and De continuo by Nicholas himself ),6 Nicholas makes use of the theory of atomic motion attributed to Mutakallimun, according to whom atoms jump from one place to another and atomic velocity is related to the amount of time an atom rests. While it is not useless to study the behaviour of individual atoms or indivisibles, Nicholas is much more interested in giving an account of generation and corruption in terms of the local motions of atoms as they aggregate and disaggregate:

Thus in the natural things there is only local movement. When this movement results in an assembly of natural bodies which gather together and require the nature of a subject, this is called generation. When they separate, it is called destruction. When through local movement atomic particles are joined to a certain subject, particles of such a kind that their arrival seems unrelated both to the movement of the subject and to what is called its natural functioning, that is called alteration.7

Nicholas could have found this theory attributed either to Democritus in Aristotle’s writings, or to the Mutakallimun in Maïmonide’s Guide for the Perplexed.8 As we shall see, the Lorain was probably infl uenced by

6 Cf. Exigit ordo, p. 219, 27, in O’Donnell, “Nicholas of Autrecourt” (The title of the work will be abbreviated as EO). All translations of Autrecourt’s work are from Universal Treatise [trans. Kennedy e.a.].

7 Universal Treatise, p. 63, EO, pp. 200, 48–201, 6: “Sic ergo in rebus naturae non est nisi motus localis; sed quando ad talem motum sequitur congregatio corporum naturalium quae colliguntur ad invicem et sortiuntur naturam unius suppositi dicitur generatio; quando segregantur, dicitur corruptio, et quando per motum localem corpora atomalia <conjunguntur> cum aliquo supposito quae sunt talia, quod nec adventus ipsorum fi eri videtur ad motum suppositi, nec ad illud quod dicitur operatio naturalis ejus, tunc dicitur alteratio.”

8 See, Moses Maimonides, Dux seu director neutrorum sive perplexorum [ Venice, 1516], p. 1a, c. 72, f. 32r: “Mundus universaliter, scilicet omne corpus quod est in eo est compositum ex partibus valde parvis, que non habent partes prae nimia sui parvitate, et nullo illarum particularum habet quantitatem ullo modo, et cum una coniuncta fuerit aliis, compositum ex eis habebit quantitatem et tunc erit corpus et si coniu-guntur duae de particulis illis, tunc utraque pars esset corpus et fi erent duo corpora secundum quosdam illorum. Omnes autem illae particulae sunt similes sibi invicem et nulla diversitas est inter illas: et dixerunt quod nullum corpus potest esse [om., ed.] nisi compositum ex istis particulis similibus positione et loco. Et secundum eos, generatio

nicholas of autrecourt’s atomistic physics 109

both texts, even if he modifi es and adapts them. It is worth noting that this claim is central to Nicholas’s physics. It allows him to develop a comprehensive account of nature that is both mechanist and reduction-ist. All natural phenomenona (even psychological phenomena) can be explained by atomic movements.9 I will try here to give an account of the principles (or elements) of Nicholas’s natural philosophy and the modes of atomic compositions.

1. Matter and Atom

The theory of atoms, in Nicholas’s mind, is a weapon against Aristotle’s theory of generation and corruption. Indeed, the atomistic conception of natural change is linked to the denial of the matter-form couple. Nicholas’s aim is fi rst to show that the matter-form distinction is a kind of metaphysical construction useless for natural philosophy.

1.1. Matter as an Atomic Fluxus

The elaboration of Ultricurian atomism relies mostly on the usual claims attributed by Aristotle to the antiqui. Nicholas never tries to identify them, as if all natural philosophy before Aristotle were atomistic. It is enough for him that the antiqui should offer a means of criticizing Aristotle. Hence, from a careful reading of the fi rst book of the Physics Nicholas retains two important notions: minima naturalia and materia prima. Adapting the former to his atomism allows him to reject the latter.

Nicholas often evokes the notion of prime matter as Aristotle’s solu-tion to the problem of generation and corruption. But Nicholas judges this alleged solution to be a mere fi ction, one that cannot claim to be better than the solution the antiqui offer. Indeed, Nicholas contends that Aristotle has no necessary argument to prove the simple generation, the passage from non-being to being:

From this conclusion, indeed, it can be concluded that the assertions of Aristotle in various places are false, and something in certain places there is only fi ction. For what he says about prime matter is neither relevant

est congregatio et corruptio segregatio neque nominant corruptionem et generationem, sed segregationem et congregationem, motum et quietem.”

9 The scope of this study will not allow to give a detailed account of Nicholas’s psychological atomism. See the fi fth chapter of my book, Croire et savoir. Les principes de la connaissance selon Nicolas d’Autrécourt, pp. 121–149.


nor true, because his basis in that investigation is that things pass from being to non-being, and vice versa. Observe that Aristotle has not at all removed the reason for the ancients’ hesitation.10

For Nicholas, the last remark is most important. Once it is demonstrated that Aristotle’s system is not evident, but only probable, it is legitimate to search for alternative solutions.11 After he examines and rejects the arguments in favour of the prime matter, Nicholas will elaborate his own conception of atomism.

A fi rst argument, called ratio aristotelis, is based on a comparison between accidental and substantial change:

But there is no necessity for <prime> matter to exist. For this necessity would result chiefl y from two arguments. The fi rst would be Aristotle’s, as it seems: A substantial change is comparable to an accidental change, but in the latter there must be something acting as subject to the termini of the change. For example, if something changes from whiteness to blackness, there is given a surface which acts as subject to both whiteness and blackness.12

If an accidental change requires a material substratum, we can conclude that substantial change (that is, generation and corruption) also requires a substratum. When Socrates’s hair whitens as he ages, Socrates is the substratum of the change. The concrete individual substance, that is the material substance, remains the same through modifi cation, and allows these accidental modifi cations. But can we conclude that there is a substratum which remains the same when Socrates passes from non-being to being? Nicholas denies it. Indeed, the only evidence for accidental change is the Aristotelian defi nition of accident as what

10 Universal Treatise, p. 68, EO, p. 204, 14–19: “Ex hac siquidem conclusione possunt concludi de dictis Aristotelis in diversis locis esse falsa, et interdum in quibusdam est solum fi ctio. Non enim habent locum nec sunt vera quae dicta sunt ab eo de materia prima quia fundamentum in illa inquisitione est quod res transeunt de esse <ad> non esse et e converso. Et videte quod nullo modo Aristoteles removit causam dubitandi antiquorum.”

11 On this topic, see Grellard, Croire et savoir . . ., chap. 4 (pp. 94–113) & 7 (pp. 192–210); Kaluza “La convenance et son rôle dans la pensée de Nicolas d’Autrécourt,” pp. 83–126.

12 Universal Treatise, pp. 48–49, EO, p. 192, 12–17: “Nunc vero non est necessarium esse materiam <primam> quia hoc esset maxime propter duas rationes quarum prima esset ratio Aristotelis ut videtur: sicut est in transmutatione accidentali, sic est in trans-mutatione substantiali; sed in illa est dare aliquid quod subicitur terminis mutationis, ut si aliquid mutatur de albedine in nigredinem est dare superfi ciem quae subicitur et albedini et nigredini.”


cannot exist by itself, but only in a subject. Even if we accept this defi ni-tion, accidental change still differs from substantial change in a signi-fi cant way:

But, admitting that in accidental change a subject is necessary, this argu-ment requires the positing of matter only because accidents, according to Aristotle, are beings only in a relative sense, so that they can have no independent existence. It does not follow from this that the same holds true in substantial generation. For Aristotle also in Book 7 of the Metaphysics, seems to mean that accidents are beings only because they belong to a being.13

Nicholas attributes the second argument in favor of substantial change to Averroes.14 It consists in a reductio ad absurdum: if we deny the existence of prime matter understood as the substratum of change, we must accept either that a thing is changed without changing, or that change rests upon on non-being. We are left with two untenable positions. Either there is no change and then we cannot explain how a thing passes from non-being to being or there is a change without prime matter, in which case there is either no substratum or, if there is, the substratum is one of the termini, that is to say the terminus a quo or the terminus ad quem. All these possibilities are absurd, Averroes contends, therefore we need to posit prime matter as a subject. Nicholas’s criticism is methodological. Averroes, Nicholas contends, presupposes as evident ( per se notum) the passing from non-being to being, but this fact is denied by those who defend the eternity of the world. Hence, prime matter is not necessary in order to explain substantial change, since the proposition “this being is changed into a substance” only means “this being exists (that is, is appearing) and didn’t exist before (that is, was not appearing).” Nicholas wants to show that atomism can give an account of this proposition in

13 Universal Treatise, p. 49, EO, p. 192, 18–22: “Sed ista ratio cogit ponere materiam quia, si in transmutatione accidentali requiratur subjectum, hoc non est nisi quia acci-dentia secundum Aristotelem sunt entia secundum quid ita quod non sunt nata existere per se; non ex hoc sequitur quod sic sit in generatione substantiae; unde idem Aristoteles in 7 Meta. Videtur intendere quod accidentia non sunt entia nisi quia entis.”

14 EO, p. 192, 23–31: “Alia ratio ad probandum materiam primam esse videtur esse Commentatoris; nam si non esset materia prima sequeretur alterum duorum, vel quod transmutatum esset sine transmutatione, vel quod transmutatio fundaretur in non ente quia vel est transmutatio, vel non est; si non sit, et certum est quod aliquid est transmutatum de non esse in esse; ergo transmutatum erit sine transmutatione; si sit transmutatio, vel ergo habet subjectum non ens, et sic alterum inconveniens; vel terminum a quo vel terminum ad quem; et utrumque est falsum quia isti sunt termini transmutationis; ergo aliquid praeter illa, et illud vocatur materia vel subjectum.”


a more economical way than the Aristotelian metaphysical construc-tion does. The only fundamentum in ente needed is the atom. Nicholas can answer both arguments by opposing the movement of atoms to the notion of prime matter. Qualitative atoms gather and separate themselves by a local movement, and there is a mutual compensation of atomic movements in the universe taken as a whole:

For in their view, when something is said to decay, there seems to be a certain withdrawal of atomic particles; when it is generated, there is a gain of others in addition. So they said that nothing decays into non-being, nor is anything generated from non-being, as is reported in Book 1 of the Physics, and Book 1 of On Generation.15

Nicholas claims to give the real meaning of the antiqui’s thesis ex nihilo nihil fi t which simply means that there is a general order in the universe such that atomic movements mutually compensate for each other. When one compound is generated, another is corrupted.16 In order to explain this fact, Nicholas uses the model of light. When light is corrupted (i.e. by night), light does not pass into the state of non-being, it only passes to the other hemisphere. The notions of being and non-being, act and potentiality, which justify the need for fi rst matter, are senseless. Nicholas explains the appearance and disappearance of objects by a strictly mechanist and reductionist theory. Finally, Nicholas accepts the thesis attributed to Democritus by Albert the Great that generation is merely the passing of an object from being hidden to being visible.17

Beyond this adhesion to a kind of Democritean atomism, Nicholas’s theory presents some original aspects. Unlike Democritus, he defends a qualitative conception of the atom as an answer to the ratio aristotelis: since Aristotle makes a comparison between accidental and substantial changes, it is enough to demonstrate that all sort of change is merely accidental. This reduction requires Nicholas to lodge a signifi cant chal-

15 Universal Treatise, p. 68, EO, p. 204, 21–24: “Quando aliquid dicitur corrumpi secun-dum eos, videtur quidam recessus corporum atomalium; quando generatur accessus etiam aliorum, et ideo dicebant quod nihil corrumpitur in non ens nec aliquid generatur ex non ente ut in 1 Phys. Et 1 De Gen. recitatur.”

16 EO, p. 193, 1–5: “Si antiqui per hanc propositionem voluerunt denotare ordinem naturalem qui est inter entia, nam quando unum ens generatur aliud corrumpitur et ita nihil generatur quin praecesserit aliquid ad quod illud quod fi t habebat ordinem naturalem in fi eri; sic intellectus eorum esset verus secundum illam opinionem.”

17 Albert the Great, De Caelo et mundo [Hossfeld], III, 2, 6, p. 233: “Et si quidam dicant eam actu esse intus, tunc ponunt latentiam formarum et generationem nihil esse nisi exitum occulti actu existentis ad apertum, sicut Democritus et Empedocles et Anaxagoras dicunt.”


lenge against Aristotle. According to Aristotle accidents cannot exist by themselves. For Nicholas accidents are verae entitates, that is to say atoms, and can exist by themselves. However they can only be perceived when compounded or combined with other atoms.

1.2. A Qualitative Atom

What is the origin of this notion of qualitative atoms? It seems that Nicholas derives this notion from the Aristotelian text itself, by assimilat-ing Democritus’s atoms with Aristotle’s minima naturalia and Anaxagoras’s homeomeron. Indeed, when Nicholas fi rst alludes to something like an atom in the Exigit ordo, he does not use the term “atoms,” but “minima naturalia,” as if Nicholas wanted to make the atomist’s most basic assumption acceptable to an Aristotelian:

For natural forms are divisible into their smallest units in such a way that these, when divided off from the whole, could not perform their proper action. And so, though they are visible when existing in the whole, they are not visible when dispersed and divided or separated. For this is true even according to the mind of Aristotle when he says that natural beings have maximum and minimum limits.18

For Aristotle, the quantity of a substantial form is determined between two limits, one minimal, the other maximal, since for any given sub-stance there are physically impossible sizes.19 Nicholas adopts this theory but gives it a new and different meaning. Aristotle does not admit the existence of discrete corpuscles and, more importantly, according to him, minima only have a potential existence. These minima cannot exist separately and have no physical existence in act.20 Nevertheless, the Aristotelian theory remains embryonic and limited. The Stagirite does not use it to explain natural phenomenona and he seems to have the refutation of Anaxagoras as its main goal. Nicholas takes advantage of this unfi nished nature of the theory. He uses the concept of minima to

18 Universal Treatise, p. 60, EO, p. 199, 42–46: “Nam formae naturales sunt ita divisibiles in minima quod seorsum divisa a toto non possent habere actionem suam et ita licet ipsa existentia in toto videantur, dispersa tamen et divisa seu segregata non videntur. Hoc enim veritatem habet etiam secundum intellectum Aristotelis dicentis: entia naturalia esse terminata ad maximum et minimum.” It’s the fi rst occurrence if we accept Kaluza’s reconstruction which puts the end of the fi rst prologue in the fi rst chapter. See Kaluza, Nicolas d’Autrécourt. Ami de la vérité, pp. 160–161.

19 See Physics, I, 4, 187b 14–21.20 See Pyle, Atomism and its Critics. From Democritus to Newton, pp. 214–217.


introduce two main features of the atom; the atom can act only inside an atomic compound and, when separated, the atom is not visible. Hence, the atom can be known only by its action inside a compound. Aristotle never gave such a property to the minima.

The assimilation of atoms to minima is not unique to Nicholas and can already be found in the writings of Albert the Great. When Albert presents the antiqui’s theories about the number and nature of the ele-ments, he curiously mixes Anaxagoras’s homeomeron and Democritus’s atoms. Indeed, he credits Democritus with the thesis according to which (1) atoms are the smallest elements of natural entities, (2) atoms are similar to these entities (atoms of fl esh or bone are similar to fl esh or bones), and (3) if we try to divide the atom, the compound will not be able to act.21 This account of atomism is interesting since it stresses the atom’s capacity to act inside a whole and the incapacity of this whole to act when atoms are subtracted (that is, when we try to divide them). This explicit assimilation of atoms to minima naturalia is an important link to understand how we can pass from the Democritean unqualita-tive atom to the Ultricurian qualitative atom.

Nicholas is not at all a Democritean when he claims that atoms are qualitative.22 Nevertheless, Nicholas is never explicit on this point. The

21 Cf. Albert the Great, De generatione et corruptione [ Hossfeld], p. 120, 44–55: “Demo-critus autem videbat quod omnia naturalia heterogenia componuntur ex similibus sicut manus ex carne et osse et huiusmodi, similia vero componuntur secundum essentiam ex minimis quae actionem formae habere possunt, licet enim non sit accipere minimum in partibus corporis, secundum quod est corpus, quod autem non accipi minus per divisio-nem, tamen est in corpore physico accipere ita parvam carnem qua si minor accipiatur, operationem carnis non perfi ciet, et hoc est minimum corpus non in eo quod corpus, sed in eo quod physicum corpus, et hoc vocavit atomus Democritus.” Minima have the role of the form according to Albert, but this problem is not solved in Nicholas’s theory. The assimilation of atoms and minima can also be red in Nicholas Bonetus, OFM, De Quantitate, in Nicholae Bonetti vir perspicassimi quattuor volumina: Metaphysicam videlicet, naturalem philosophiam, praedicamenta, necnon theologiam naturalem [Venise, 1505] f. 81rb: “Respondet tibi Democritus quod non, sed omne continuum actu fi nitum potest resolvi usque ad indivisibilia simpliciter, et in hoc videtur concordare Aristoteles de continuo naturali quoniam in primo physicorum 3 commento et 2 de anima 42 dixit quod est devenire usque ad minimam carnem.”

22 The other ancient atomism known to medieval philosophers, the Epicurean one, also claims that atoms have no quality. See Cicero, De natura deorum, [Van den Bruwaene], vol. 2, II, 94, p. 119: “Isti autem quemadmodum adserverant ex corpusculis non colore, non qualitate aliqua quam poiotèta Graeci vocant, non sensu praeditis sed concurrentibus temere atque casu mundum esse perfectum, vel innumerabiles potius in omni puncto temporis alios nasci, aliso interire. Quod si mundum effi cere potest concursus atomorum cur porticum, cur templum, cur domum, cur urbem non potest quae sunt minus operosa et multo quidem faciliora.”


fi rst time he deals with this question, he simply asserts that atoms have different natures:

If the question is raised whether the atoms are of the same kind or are of different kinds one must say, of differents kinds. But the means of proving the different kinds will perhaps become apparent later.23

Unfortunately, Nicholas never fully explains what this means. Only once he repeats his assertion without offering further details.24 On at least two occasions, he explicitly describes two varieties of qualitative atoms: white atoms and fi re atoms.25 Moreover, when dealing once again with the question of atomic differences, Nicholas provides some precious interpretative keys:

It should be observed that the eternity of things could be understood in one of two ways. One is that they would always remain integral as some composite whole, just as they are now. For example, in Socrates there are many realities of different natures joined together, such as fl esh, bone, and soul.26

23 Universal Treatise, p. 69, EO, p. 205, 15–20: “Si autem quaeratur de illis atomali-bus, an sint unius rationis vel alterius, dicendum quod alterius; sed ex quibus prob-etur diversitas rationum inferius forsitan apparebit.” We may assume that atoms by themselves have no quality but when put in the compound, they change their nature and become active.

24 Cf. EO, p. 251, 5–8: “Unde de qualibet re non erit nisi punctus et sicut torchia fi t non nisi sicut aggregatum ex pluribus diversarum rationum, sic lumen est aliquid aggregatum ex pluribus diversarum rationum.” We must admit that it is hardly imagin-able that light is compounded from atoms different by nature. This text would attest a Democritean atomism. But the following is not at all Democritean.

25 Cf. EO, p. 189, 26–29: “Sic hic habeo media satis probabilis ad concludendum quod conclusio de aeternitate rerum est probabile, sed quia non possum ostendere illas modiculas albedines ad modum granorum ire et venire, aliqui forsan discrederent; non tamen propter hoc est negandum.”; p. 259, 3–8: “Propter quod sciendum est quod ut innui aliqualiter supra virtus calefactiva perfectissima copulatur quantum ad operari cum igne perfectissimo, ita quod operatur ubi est ignis perfectissimus, si sit materia disposita quam corpora ignita subintrant, et quanto magis subintrant, ei penetrant, supposita identitate materiae, tanto virtus calefactiva perfectior operatur.”

26 Universal Treatise, p. 141 (translation modifi ed), EO, p. 251, 44–48: “Et adver-tendum quod secundum alterum duorum posset intelligi aeternitas rerum vel quod remanerent semper sub integritate alicujus totius copulati sicut nunc sunt, ut verbi gratia, in Socrate sunt multae realitates diversarum rationum copulatae ut caro, os et anima”. The second part of the alternative opened by the ‘vel’ is the following, p. 252, 6–10: “Alius esset modus intelligendi aeternitatem in rebus secundum viam segregationis ut nullicubi esset Socrates per modum totius copulati, sed alicubi esset albedo ejus causata, alibi ejus virtus et sic de aliis, et tandem revoluto circulo magni orbis, iterum fi eret congregatio; et horum hororum recipe probabiliorem”. The problem under discussion here is about eternity. See Kaluza, “La récompense dans les cieux. Remarques sur l’eschatologie de Nicolas d’Autrécourt,” p. 90, particularly the note 15 which deals with these questions.


We already know that the soul is an atom. This was also the case for Democritus.27 What is new is the existence of atoms of fl esh and bone. In other words, Nicholas identifi es atoms with homeomeron, as we have already said, and this offers a key for understanding the shift toward a qualitative atomism.

We have seen how Albert in his commentary on De Generatione et corruptione dealt with atoms of fl esh and bone as little particles of fl esh and bone ( parva caro) and this presentation of Democritus’s ideas no doubt prepared the way for the Ultricurian confusions.28 Nicholas even uses the same example, attributed by Albert to Democritus, of atoms of fl esh and bones:

This is true of a simple effect but not of one composed of things of dif-ferent natures. In a complex effect, it would be otherwise, because then in truth there are different beings there, as in Socrates there are bones, fl esh, soul, blood, etc.; and so one must posit different causes there.29

Nicholas’s equivocal notion of thing (res) complicates and undermines this text. But here, we may assume that Nicholas is dealing with atoms and compounds because of the use of being (entia). Of course, atoms are the real beings (vera entia).30 Similarly, this qualitative conception of the atom could have been infl uenced partly by the Mutakallimun, who discussed atoms in terms of substance and accident. The Mutakallimun even claim that atoms of snow are white. Something similar crops up in the Ultricurian classifi cation of atoms.31

27 For Democritus, see Aristotle, De anima, I, 2, 404 a 1–10. 28 See the quotation in the note 21.29 Universal Treatise, p. 148 (translation modifi ed), EO, p. 256, 21–24: “Et hoc est

verum de effectu simplici, non de composito ex rebus diversarum naturarum, secus esset in alio, quia tunc quantum ad veritatem ibi sunt diversa entia sicut in Socrate ossa, caro, anima, sanguis etc.; et ideo ibi oportet ponere diversas causas.”

30 Cf. EO, 225, 43–46: “Sic hic verum est quod omnis entitas vera, quae in me est nunc, semper fuit et semper erit, sed non erunt secundum indistinctionem subjectivam ut nunc sunt.” On this equivocal notion see Kaluza, “L’éternité . . .,” pp. 234–238, “La récompense dans les cieux . . .,” p. 90, n. 15. Also, Dutton “Nicholas of Autrecourt and William of Ockham on Atomism, Nominalism and the Ontology of Motion,” p. 66, n. 8.

31 Moses Maimonides, Dux seu director . . . [ Venice, 1516 ], f. 32v: “Sed tale accidens, secundum ipsos, invenitur in quolibet atomorum ex quibus componitur illud corpus, verbi gratia, albedo partis nivis non invenitur in suo universo solummodo sed in qualibet substantiarum illius nivis est albedo et idcirco invenitur albedo in composito ex illis.” There is nevertheless an important difference since for Mutakallimun atoms are substances which can have accidents. Nicholas tries to go beyond the accident-substance dichotomy.


It seems clear that Nicholas’s atomism draws from both Democritus, as transmitted and deformed by the Aristotelian tradition, and from the Mutakallimun. It is important to note, however, that Nicholas’s goal is to build a consistent atomistic theory and not simply to set Democritus against Aristotle. He is looking for the best explanation of natural phe-nomenona and he believes that the best explanation is atomistic. And this means that he must go beyond the few clues left in the writings of the antiqui in order to complete his theory.

1.3. The Unity of Atomic Flow: Principles of Atomic Composition

The interpretation of matter as an atomic fl ow is tied to the critique of the metaphysical notion of form and matter, or act and potentiality. By rejecting prime matter as a substratum, Nicholas is also rejecting the idea that the form is an actualization of matter and, a fortiori, he is rejecting the idea of substantial form. The problem Nicholas then faces is to explain what gives unity to atomic compounds. Since Nicholas admits the possibility of monsters in nature, understood as inconsistent (disconveniens) atomic compounds,32 he must provide an account of what produces consistent compounds. The answer is two-fold. On the one hand, Nicholas defends an atomistic mereology that relies on the par-ticular nature of different atoms. On the other hand, his answer relies on the astral causality that determines the atomic composition.

Since accidents are the result of corpuscles, it is necessary to fi nd a way of allowing a distinction between accidental and essential proper-ties. Accidents are atoms which superfi cially modify the compound from which they can be subtracted. Essential properties are necessary to the existence of the compound. In other words, after giving an atomistic account of matter, Nicholas must give the same account of form. Hence, he introduces a distinction between atoms which are essential to the whole, such that their separation leads to the destruction of the compound, and atoms which separation does not alter the nature of the compound. Atoms of the fi rst kind allow an operation or a movement (as is the case for the soul).33 Atoms of the second kind are nothing but qualities like whiteness:

32 EO, p. 205, 43–44: “In rebus materialibus extra, propter disconvenientiam <et convenientiam> in congregatione dicuntur interdum monstra interdum bene composita.”

33 On the atomic status of the soul, see Democritus, quoted in Aristotle’s De anima, I, 2 (see note 27) and the Mutakallimun, quoted by Moses Maimonides, Dux seu


On the basis of the foregoing discussion it might be said that these accidents are only certain atomic particles, and that they are not in the subject except as a part in the whole, but a part, one must understand, that is essential and necessary to the whole. Still more can these parts be sayed about the substance of the subject. Upon the departure of these atoms in the substance, what is called the functioning of the thing, and the movement which previously appeared in the thing, cease to appear. Upon the departure of the other atoms they do not go away. These ought more properly to be called accidentals of the subject; yet they too are as a part in the whole.34

Among atoms which are parts of a whole, we have to make a distinc-tion between necessary parts, such that the whole cannot exist without them, and accidental parts, such that the whole can exist without them. Since each atom is qualitative, this distinction is similar to the one between essential and accidental properties. Essential properties allow the compound to operate. Generation is explained by the gathering of such atoms, and alteration by the separation of atoms of the second kind. With this distinction, Nicholas claims to be able to dissolve all the false Aristotelian problems linked to the notions of quiddity, form, and so on.35

But it might seem odd that Nicholas still uses the notion of sup-positum at all. What does this Aristotelian notion mean in an atomistic context? Nicholas is not very clear on this point and, at fi rst glance, it seems that he should give up such concepts as species and genus. On at least one occasion, he explains what an atomistic conception of form and species might mean in the case of human beings. After reducing generation and corruption to local movement, he writes:

Perhaps there is something there which connects and retains the indi-visibles in this union, as a magnet does with iron. The stronger the force

director . . . [ Venice, 1516], f. 32v: “Quidam autem eorum dicunt quod anima componitur ex atomis quae sine dubio conveniunt in accidente per quod sunt anima et quidam dixerunt quod ipse virtutes et ipse substantie coniuguntur cum substantiis corporeis et non evadunt quin rationem anime ponant accidens.”

34 Universal Treatise, pp. 68–69, EO p. 204, 38–45: “Secundum praedicta diceretur quod talia accidentia non sunt nisi quaedam corpora atomalia, nec sunt in supposito nisi sicut pars in toto; verumtamen intelligendum quod est quaedam pars essentialis et necessaria toti; et istae magis possunt dici de substantia suppositi. Et ista sunt atomalia quibus abeuntibus non apparet amplius illud quod dicitur operatio rei, nec motus qui prius appareret in re; alia sunt quibus abeuntibus non abeunt; ista magis debent dici accidentalia suppositi; sic tamen sunt sicut pars in toto.” This text is examined by Kaluza, “Les catégories dans l’Exigit ordo de Nicolas d’Autrécourt,” pp. 101–102.

35 EO, p. 204.


of this thing, the longer the subject survives as a subject. If there were a force of this kind, it would be called the quasi-formal principle of the thing.36

When local movement leads to the gathering of mutually consistent atoms, they produce a suppositum which, according to Nicholas, simply means a totality of essential atoms. Nicholas then adds that one of these essential atoms plays the role of a magnet and attracts the other atoms. The gathering of the atoms in the compound depends on the strength of this atom, which he refers to as a “quasi-formal principle.” When it ceases to act, the other atoms spread out the whole.

We may assume that this atom is not the only vital principle in liv-ing beings, since the death of animals is not the dispersion of atoms, but the departure of some essential parts. Nicholas is more probably dealing with the problem of decomposition. In each compound, there is an atom whose function is to attract atoms and keep them together. As long as it continues to operate, the compound exists as a perceptible whole. When it stops operating, the compound is disintegrated. The fi fth proposition of the Mutakallimun, as reported by Maimonides, could well have infl uenced Nicholas here.37 The Ultricurian innovation is the link between this formal atom and an astral causality. Indeed, the last level of Ultricurian atomism relies on the interaction between lunar and sublunary worlds.

2. From Atoms to World: The Nature of Atomic Compound

Nicholas still must explain how his atomism provides a convincing account of natural phenomenona. More specifi cally, the challenge he faces is to explain how consistent atomic compositions can be possible.

36 Universal Treatise, p. 63, EO, p. 201, 9–11: “Et forsan sicut adamas ferrum, ita est ibi unum quod connectit et retinet in tali colligatione ipsa indivisibilia, et secundum hoc quod est majoris vigoris magis durat illud suppositum in ratione suppositi; et illud, si sic esset, diceretur quasi principium formale rei.”

37 Moses Maimonides, Dux seu director . . . [ Venice, 1516], f. 32v: “Idem dixerunt in corpore mobili quod quaelibet substantiarum suarum separatarum movetur et idcirco totum movetur. Similiter secundum ipsos vita invenitur in qualibet atomo vivi, sic et sensus invenitur in qualibet substantia separata sentientis, quia sensus et intellectus et scientia secundum ipsos sunt accidentia sicut albedo et nigredo, sicut explanavimus in eorum opinio sed obiectum fuit eis in anima. Convenerunt autem in hoc quod est accidens inventum in una substantia separata de universitate substantiarum ex quibus componitur homme et illa universitas dicitur habere anima qua substantia separata est intra ipsam universitatem.”


As we have seen, Nicholas reduces all change to local movement, but what are the principles behind local movement and what is the role of matter?

2.1. Matter, Disposition and Virtus

A constant thesis in the Exigit ordo is that matter, understood as an atomic fl ow, is designed to prepare or to receive a formal atom called virtus which is an essential part. This atom constitutes the principle of unity and operation of the compound, in other words its form.

Nicholas gives a clear example of his theory when he discusses com-bustion.38 How is fi re produced? It needs a virtus ignitiva, in addition to the heat that is produced by a virtus calefactiva. If one wants to escape the standard criticisms of virtues, summed up in Molière’s joke about the “vertu dormitive de l’opium,” it is necessary to explain how they operate.39 While these virtues or forces are a particular kind of atoms, they never act alone. According to Autrecourt, these virtues are trig-gered by superior agents, the celestial bodies. Atomic action requires a copulatio between the star and the atom. The force found in the atomic compound is a formal effect of the same force found in the appropriate star.40 Light can heat because the atoms which compose it follow the movements of the stars. Hence, the celestial body is an effi cient cause that acts upon an atom which is the formal cause in the compound. Still this is not enough. How can we explain why a bellows, when used gently, can stir up a fi re, while using it to create a vigourous wind might well extinguish the fi re? What is the role of wind in the generation of fi re? According to Nicholas, it provides a kind of help. When the wind is gentle, it expels from the compound atoms liable to delay combustion (that is, atoms that would delay the copulatio between the atomic and celestial virtues). Conversely, a strong breeze will expel atoms liable to facilitate the copulatio, those hot atoms which are the condition for

38 EO, p. 257, 3–9: “Potest tamen dici quod illa virtus inexistit alicui superiori agenti et haec virtus habet quamdam copulationem in operari cum effectu formali, ita quod quando approximatur ignis alicui calefactibili, saltem <si> ignis talis sit perfectus, tunc calefacit illud calefactibile, et secundum hoc quod est calefactibilius, copulatur sibi virtus calefactiva perfectior; et secundum hoc est falsum dicere quod lumen calefacit vel quod motus nisi quia ista assequuntur virtutes quaedam calefactivae.”

39 For a contemporary outlook on this topics, see Gnassounou & Kistler (eds.), Les dispositions en philosophie et en science.

40 For a more detailed account of the celestial causality, see Grellard “La causalité chez Nicolas d’Autrécourt.”


the forces to act. These atoms, it would seem, dispose or prepare the composite towards certain behaviors, in other words, they are similar to material causes.41

From this example, Autrecourt applies the model of a copulatio be-tween stars and atomic virtues to different natural phenomenona, and fi rst of all to generation and corruption. Hence, he rejects spontaneous generation which he assimilates to normal generation. In both cases, we have an equivocal generation, that is, the reception of atomic vir-tues into a prepared matter. The atoms of a worm can be received in a putrefi ed matter; similarly, the sperm of a man or a donkey is not the cause of the generation of another man or donkey but only a requisite part. Sperm is the material condition, the atomic virtus in the man is the formal cause and this formal cause still requires the effi cient action of the star.42 Usually, if these material and formal conditions are lacking, the star cannot act on the compound. Nevertheless, we may

41 EO, p. 257, 9–14: “Ventus autem facere videtur ad generationem ignis ut in sibillo oris vel in sibillo artifi ciali, praecipue quando est moderatus; et hoc est quia cum tali vento recedunt aliqua corpora quae impediebant receptionem ignis intali subjecto, ita quod virtus ignitiva perfecta non copulabatur sibi; quando vero est immoderatus, tunc etiam removentur illa quae disponebant subjectum ad igneibilitatem.”

42 EO, pp. 257–258: “Ex illis regulis suprapositis quod unus effectus non precedit nisi ab una causa etc., sequitur quod anima unius hominis qualiscumque sit non est producta ab alio homine quia seqitur: ab eo producta, ergo praecise producta; igitur alterum non est ignobilius quia causa numquam potest excedere suam perfectionem in producendo ut dictum est; neque aeque perfecta quia, ut dictum est supra non est dare duos effectus aequales in una specie; igitur erit perfectior; et hoc est falsum, immo plerumque imperfectior reperitur quantum ad omnes virtutes et operationes. Producitur igitur ab aliquo agente superiori; non dico ab agente perfectissimo nisi sit effectus perfectissimus ut probatum est superius. Et si homo, qui diceretur producens sit perfectior, hoc accidit qua tantum videtur facere imperfectior circa generationem hominis sicut magis perfectus. Et sicut dictum est de homine sic intelligendum de asino et omnibus aliis, et per consequens nulla erit productio univoca quia vel causa esset imperfectior quod non potest esse, vel aeque perfecta quod improbatum est supra, vel perfectior, et hoc accidit sibi in quantum causa effectus ut statim dictum est. Homo tamen bene potest esse aliquorum praerequisitorum ad quae forma hominis habet ordinem ut seminis et aliorum; homo, id est, aliqua virtus inexistens homini. Et con-siderandum est juxta praedicta quod illa est falsa, quod virtus aliqua producit effectum perfectiorem in subjecto perfectiori et imperfectiorem in imperfectiori quia virtus aliqua non habet <nisi> unum effectum ut dictum est supra, et ideo vel illum producit vel nullum. Sed verum est quod, nisi subjectum sit conveniens ad recipiendum illum, non producet in illo subjecto, sed cum illo subjecto copulabitur in operari virtus imperfec-tior. Subjectum enim indispositum nihil immutat de natura agentis quin, si producat, producat secundum convenientiam suae naturae.” On the sperm as a preparation, see the quotation in the following note. For a different account of the role of the sperm, according to the medical theories, see Jacquart, “L’infl uence des astres sur le corps humain chez Pietro d’Abano”.


assume (even if Nicholas does not explicitly say so) that monsters are produced when one of the material or formal requisites are partially lacking in such a way that the stars are still capable of acting on the composite, but the result will not be normal. Hence, all atomic changes are the result of a celestial effi ciency into a formal or accidental atom in a preexistent atomic fl ow, materially disposed to receive this atom. However, within this celestial causality, material conditions have a role to play in defi ning the resulting species. At fi rst glance, Nicholas seems to assume that such a notion is useless. He fi rst underlines that the trans-temporal identity of the same individual is never complete. The young Socrates and the old one are not exactly the same individual. This person is continuously modifi ed by the movements of the atoms, and is never identical to himself between moment t and moment t+1. If a child could become an old man instantaneously, we would not be able to identify him as the same person. But since these changes are continuous and since the general order of the atoms remains the same, we can speak about these various atomic compositions as the same person.43 From this, we may conclude that according to Nicholas all sorts of form, even the substantial form, are reduced to the fi gures

43 EO, pp. 251–253: “Nunc posset sic intelligi quod Socrates sic esset aeternus quod sic semper esset sicut nunc est, sic intelligendo quod, cum Socrates non sit omnino idem sibi puer et senex, immo aliquo modo variatu de hora in horam, intelligetur quod quando desineret esse sub una dispositione utpote sub dispositione pueritiae, alibi esse sub eadem dispositione et postea alibi et sic semper usque ad circulum, donec fuisset ubique et non solum secundum horas vel dies, sed etiam secundum momenta, ut statim cum hic desineret esse sub una dispositione, inciperet esse alibi sub eadem. (. . .). Et ex praedictis potest dici, si quaeratur an puer sit homo, si appellatio recipiatur a fi gura magis in genere, tunc quia habet fi guram ad modum hominis, caput sursum, pedes deorsum, ut sic deberet dici homo. Sed appellationem ex hoc recipere non est conveniens, sed magis ex operatione et virtute inexistente; nunc sicut non habet operationem, ita posset dici non habere virtutem et solum deberet dici quod habet virtutem ut operatur. Et ita de puero unius diei, si moriatur, potest dici quod nec habuit virtutem ridendi neque ratiocinandi, licet bene habuerit aliquas praeparationes remotas ad hoc sicut est in spermate. Consuetum tamen est dici quod homines et <illi sunt> ejusdem speciei; et causa consuetudinis forsan est ut homines magis compatiantur eis. Et si puer non habet virtutes hominis, conveniens est inquirere an virtus existit suo quiete, verbi gratia quando aliquis non movet lapidem vel aliquod aliud grave, utrum habeat virtutem movendi lapidem. Videtur quod non, quia tunc ejas esse esset otiosum pro illo tempore. Videtur contra, quia tunc nullus deberet eligere quietem nec virtus naturalis cum fatigat naturam, natura non deberet naturaliter ad hoc inclinari, cum in hoc consisteret ejus destructio; et ideo posset satis dici quod remanet sub quiete, nec est otiosum cum in hoc accidit conservatio ejus in supposito, alias causaretur alibi et hic desineret esse et loco ejus esset imperfectior. Unde corpus caeleste cum quibusdam concurrentibus spritibus causat talem virtutem qui continuatio opere utpote motionis gravis recedunt, et loco eorum veniunt magis imperfecti.”


or to the general appearances of a thing. If somebody has a head at the top and two feet at the bottom, we can call him a man. Nicholas, however, concedes that such a conception of form is not enough. Rather, he defi nes form as an operation or virtus (that is an atom) which allows us to classify individuals into species and genus. Moreover, it is not only this operation that allows us to classify individuals into species, but also the tendency of these types of atoms to act in certain ways. Otherwise, a stillborn child could not be called a human being, since he would lack the ability to laugh and reason. If we do not want to call it a human being out of mere custom, we have to assume that the appropriate virtue is already present, if latent, inside the material preparation (the sperm) and that this virtue is just waiting to be moved by an astral causality.

We can see that from an external point of view, atomic movements depend upon astral movements. But there is also an internal condition to be considerered, the inter-atomic void.

2.2. Void as a Condition of the Atomic Movements

The treatise De vacuo is one of the best known sections of the Exigit ordo.44 Against Aristotle, Autrecourt argues for the possibility of the void. Not a separate void, but an inter-particulate void, that is, a void internal to the compound. Aristotle attributes this doctrine to Democritus in the fourth book of the De Caelo. Nicholas uses it mostly to give an account of the phenomenon of condensation and rarefaction.

It is well known that Nicholas, like the Mutakallimun,45 explains atomic movements by the motion of an atomic body through atomic places. In De vacuo, Nicholas demonstrates that these atomic places are empty. Contrary to the scholastic doxa, motion in a void does not

44 See specifi cally, Grant, “The Arguments of Nicholas of Autrecourt for the Exist-ence of Interparticulate Vacua,” pp. 65–69; Kaluza, “La convenance et son rôle,” pp. 100–103; Grellard, Croire et Savoir . . ., pp. 211–218.

45 Moses Maimonides, Dux seu director, f. 32r: “Dixerunt quod motus est mutatio substantiae separatae de numero atomorum unius scilicet singularis substantiae ad aliam substantiam propinquam et ex hoc sequitur quod non est unus motus velocior alio. Et secundum hanc positionem dixerunt quod vides duo mobilia pervenire ad duos terminos diversos in remotione in eodem tempore, non est quia unus motus est velocior alio sed causa eius est quia mobile cuius motus dicitur tardior habet plures quietes in spatio suo quam illud quod dicitur velocius. Et ideo dixerunt in sagita fortiter ab arcu emissa quod sunt quietes in spatio motus ipsius.” See Murdoch, “Atomism and Motion in the Fourteenth Century;” Grellard, “Nicolas d’Autrécourt.”


imply instantaneous velocity since every movement is a relative one, and depends on the time during which each atom rests. Moreover, each atom moves place after place, by a local movement. From these assump-tions, Nicholas explains condensation, that is, the quantitative alteration of bodies, by the local movement of atoms inside the compound. The classical example of the rarefaction of new wine is explained not by the generation of a new quality (which would mean the arrival of new corpuscles), but by the separation of parts already present in the body and the concomitant introduction of a void between these atoms. In the same way, water seems denser than air because there is more void between its parts. Hence, in the case of liquid and gas, the mutual proximity of the parts give an account of density, and their distance explains the rarity.46 In the case of solid bodies, Nicholas offers two different explanations: alteration is caused by an increase or decrease of the void, or by the arrival or the departure of new atoms.47

Against Aristotle, Nicholas asserts that nothing is absolutely heavy or light. These concepts are relative and depend upon the inter-particulate void and the proximity of atoms to one another. Atoms can move within a composite because of the presence of an inter-particulate void, other-wise two bodies could be in the same place at the same time, which is universally believed to be false.48 This is how the antiqui explained the possibility of a local increase of a body. How else can we explain, for example, how our limbs grow in size, if not by the addition of atoms? Nicholas gives the following explanation of this phenomenon. Let us assume that a body is composed of four atoms, abcd. These four atoms are indivisible. Each of them has a specifi c quantity since each has a situalitas and they are mutually apart from one another. Let us assume also that between each atom there is an empty space.49 Body abcd will grow as additional atoms come to fi ll in spaces that grow between abcd. For example, atom e will insert itself between a and b to produce a new body, aebcd. The ability to receive new atoms depends on a kind of dilation which creates empty spaces within the composite. But there is no modifi cation of the atoms themselves, that is, no modifi cation of

46 EO, p. 218, 11–31. 47 EO, p. 217, 33–37: “Nam non dicimus quod densum sit per generationem alicujus

novae qualitatis quae prius non erat, sed solum est densum per recessum corporum ut in lana, vel quia partes coeunt, id est, quod magis propinque se habent quam prius. Et rarum non erit nisi quia partes illius corporis magis distant quam prius, et ita densum vel rarum non est nisi per solum motum localem partium.”

48 EO, p. 221, 42–222, 7. 49 EO, p. 208, 10–18.


their own properties and quantity. The compound appears to grow because it comes to be made up of additional atoms, atoms that, on their own, are imperceptible.50

Nicholas may have John Buridan’s argument against atoms in mind. In his Questiones super De Caelo, Buridan wants to prove that an atom must be heavy and, therefore, must be divisible.51 His purpose is to show that any quantifi ed body is liable to increase or decrease. If we can show that an atom is a body which can sometimes be bigger and heavier, and sometimes smaller and lighter, Buridan believes he can prove his point: an atom possesses quantity and is divisible. On this point, Buridan refutes an adversary who claims that elements (water, air, earth, fi re) are composed of atoms of different weights, such that atoms of air are less heavy than atoms of earth. Buridan begins with an Aristotelian argument. Imagine a body composed of three atoms abc. If this body condenses, it will be smaller, but it will still be composed of these same three atoms. We must conclude that the individual atoms have lost weight. Secondly, if we examine the status of quantitative external parts outside each other, and if we assume that an earthly body is composed of three atoms abc, to which we add a fourth similar part d, we must claim that abcd is more extended and heavier than abc. Thirdly, if we examine the phenomenon of calefaction, we are faced with three assumptions: either, 1.) calefaction is produced by the com-pression or condensation of the parts such as they become closer to each other; 2.) calefaction is produced by the addition of a qualitative degree of hotness (as in the theory of intension and remission of forms); or, 3.) calefaction is produced by the addition of an atom of hotness, that is, a quantitative part. Against this last possibility (the other two do not interest him in this case), Buridan claims that a hotter body should be heavier and more extended. This thesis of the addition of hot corpuscles

50 EO, p. 221, 12–24: “Pone corpus aliquod compositum est ex indivisibilibus quattuor a b c d; dico quod inter ista indivisibilia sunt vacuitates; perpone inter a et b per unum indivisibile; dico quod ad adventum rei convenientis quasi dilitabant se, ita quod quodammodo elongabant se, puta a b per vacuitatem duorum indivisibilium, et ibi recipient [se] intra se illud quod est conveniens suae naturae, et sic fi et augmen-tatio. Et intellectus hujusmodi propositionis illius: quaelibet pars aucti est aucta, non comprehendit omnino partes primas corporis aucti ut sit intellectus quod indivisibile augetur; sed intellectus est quod post augmentationem de qualibet parte composita demonstrabili ad sensum comparata ad talem, quae antecedebat augmentum, verum est dicere quod est major, ut puta manus post augmentum est major quam ante aug-mentum, et sic in aliis.”

51 John Buridan, Expositio et Quaestiones in Aristotelis De Caelo [Patar], L. III, q. 1, pp. 513–518.


is indeed Nicholas’s thesis.52 But for Nicholas, the link between exten-sion and heaviness is not necessary as in Buridan’s argument, because Nicholas contends that a body can be more extended and lighter, if it is composed of more empty places. Hence, the arrival of atoms of hotness neither increases the composite’s extension nor its weight. Likewise, the modifi cation of a body’s size does not imply any modifi cation of the atoms, but only of the empty places which separate them.

3. Conclusion

From the celestial bodies, which simplicity renders them eternal, to the sublunary world composed of atoms, Nicholas offers a completely atomized cosmos. In his attempt to offer a non-Aristotelian answer to the twin problems of eternity of the world and of generation and cor-ruption, Nicholas constructs an atomistic theory using parts he could have drawn from Aristotle’s own writings in natural philosophy and from the Mutakallimun. He then attempts to elaborate a non-metaphysical physics founded on very few principles: qualitative atoms, local motion, and celestial causality. In this context, atomic fl ows provide an account of the make-up of these compounds and the possible changes they may undergo in the sublunary world. But at the same time, Nicholas’s atomism testifi es to the diffi culty that medieval natural philosophers had to face when they sought to escape Aristotle’s most basic frameworks. Indeed, by attempting to rewrite Aristotle’s theses in atomistic terms, Nicholas effectively preserves Aristotle’s framework, a framework that would remain essentially unchallenged for at least another 200 years. Due to the condemnation of 1347, it is not possible to know how Nicholas may have contributed to the rejection of the Aristotelian model. But Nicholas’s ingeniousness in the development of an alterna-tive physics deserves, in the words of Edward Grant ,“a tribute to his courage and intellectual acumen.”53

52 EO, p. 257: “Supra in tractatu de aeternitate rerum dixi quod quando ignis dicitur produci quantum ad veritatem non est nisi adventus aliquorum corporum calidorum, et frigidditas per recessum illorum et adventum corporum frigidorum ut in aqua calida quae sibi derelicta revertitur ad naturam priorem, scilicet frigiditatem, per recessum calidorum corporum et accessum frigidorum.” On the Ultricurian approach of intension of forms, see Grellard, “L’usage des nouveaux langages d’analyse dans la Quaestio de Nicolas d’Autrécourt. Contribution à la théorie autrécurienne de la connaissance.”

53 Grant, “The Arguments . . .,” p. 68.

WILLIAM CRATHORN’S MEREOTOPOLOGICAL ATOMISM

Aurélien Robert

Little is known about Crathorn’s life and career, except that he was a Dominican friar who lectured on Peter Lombard’s Sentences in Oxford around 1330–32—his only surviving work.1 He’s often considered by recent scholars as a secondary witness to the most important debates in fourteenth-century philosophy, but rarely as an inventive thinker. None-theless, isolated arguments have been carefully examined, notably his solution to scepticism2 and his criticism of William of Ockham’s theory of mental language.3 But, concerning the question of the existence of indivisibles in a continuum, he’s generally mentioned as a simple doxographer, even if John E. Murdoch recognized him as a defender of an original theory of speed in such a context.4 Contrary to this common reading, we would like to show how singular and interesting is Crathorn’s atomist position, which can neither be reduced to that of his Oxonian predecessors Henry of Harclay and Walter Chatton, nor identifi ed with his Parisian contemporaries Gerard of Odo, Nicholas Bonet and Nicholas of Autrecourt.

Indeed, Crathorn doesn’t limit himself to mathematical analysis of the divisibility of a continuum, but puts forth the foundations of a genuine atomist physics. In this theory, an indivisible is not conceived as a purely mathematical point anymore, but rather as the ultimate component of reality, from which a conception of motion rivalling the one defended by Aristotle in his Physics can be derived. At the heart of Crathorn’s physics, the indivisible thus acquires a new ontological status: it is a thing (res), actually existing (existens in actu) and not only

1 The Questions on the book of the Sentences have been edited by Hoffmann, Quaestionen zum ersten Sentenzenbuch. For biographical elements, cf. Schepers, “Holkot contra dicta Crathorn: I. Quellenkritik und biographische Auswertung der Bakkalaureatsschriften zweier Oxforder Dominikaner des XIV. Jahrhunderts;” Courtenay, Adam Wodeham: An Introduction to his Life and Writings. For a general survey, see our “William Crathorn.”

2 Cf. Pasnau, Theories of cognition in the Later Middle Ages.3 Cf. Panaccio, “Le langage mental en discussion: 1320–1335;” Perler, “Crathorn

on Mental Language.”4 For example, Murdoch, “Atomism and Motion in the Fourteenth Century.”

128 aurélien robert

potentially, which possesses a kind of extension (quantitas dimensiva) and a certain nature or perfection (quantitas perfectiva). Moreover, much of Crathorn’s effort was directed towards demonstrating that the number of indivisibles in nature is fi nite, if not countable. Therefore, as he will conclude, the indivisibles are real parts of a continuum.

Our aim in this paper is to show how Crathorn brought out an impor-tant turn in his natural philosophy, from the indivisible considered as a mathematical point to the atom conceived as a physical entity. Needless to say that he isn’t the only one who endeavoured to conceptualize the nature of indivisibles in a more physicalist fashion, but in some respect he’s probably one of the most systematic atomistic thinker of the fi rst half of the fourteenth century even if, as we shall see, some theoretical hesitations still persist in his view.

To understand Crathorn’s originality, it should be remembered that his ontological analysis of the indivisible is based on two distinctive features: it depends on a special conception of the part-whole relation; and it is supported by a systematic use of the notions of place (locus) and position (situs). For this reason, we may call Crathorn’s theory a “mereotopological atomism”.5 The notion of place is so central that it will serve as a tool for reconstructing the physical nature of atoms, that is to say their quantity (dimension and perfection); for, as we shall see, indivisibles are primarily defi ned by the place they occupy in the world. Consequently, this mereotopological strategy will also allow him to avoid falling into some of the traditional blind alleys of indivisibilism, e.g. the explanation of the composition of a quantity from extensionless points and the problem of contact between indivisibles.

Before going through Crathorn’s arguments, we must mention how important and recurrent is the topic of indivisibilism in the Questions on the Sentences.

5 We borrow the term “mereotopology” from contemporary science and phi-losophy, using it in a slightly different way. For a contemporary defi nition, see Smith, “Mereotopology: A Theory of Parts and Boundaries,” p. 287: “Mereotopology . . . is built up out of mereology together with a topological component, thereby allowing the formulation of ontological laws pertaining to the boundaries and interiors of wholes, to relation of contact and connectedness, to the concepts of surface, point, neighbourhood and so on.”

william crathorn’s mereotopological atomism 129

1. The Importance of the Continuum Question in Crathorn’s Writings

The most signifi cant occurrence of our topic is to be found in q. 3 of the Questions on the Sentences after a long discussion about the nature of natural knowledge and language (q. 1 and q. 2). As a consequence, one shouldn’t be surprised by the gnoseological pretext of this fi rst long digression: does the wayfarer understand cum continuo et tempore.6 The few commentators who have studied the continuum question in Crathorn have usually restricted their reading to this sole passage, but many others are relevant in the Quaestiones.

As early as q. 1, Crathorn adopts atomist explanations of natural phenomena as the diffusion of light when explaining the multiplication of species through a medium.7 In q. 4, when questioning the possibility of knowing God’s infi nity, our topic reappears in the explanation of the infi nite.8 But the most important passages on atoms and continua can be traced in Crathorn’s discussion of Aristotle’s categories, especially in his long developments in q. 14 on quantity, and in q. 15, entirely dedicated to the quantity of indivisibles.9 Finally, after developing his own interpretation of Aristotelian categories, Crathorn turns to the nature of time and motion in q. 16,10 where he defi nes more precisely the central elements of what could be the foundations of an atomist physics. Therefore, for a complete reconstruction of Crathorn’s theory, one must at least take into account all these passages in which indi-visibilism is at stake.

To be absolutely exhaustive, one should also look at the series of forty-two quodlibetal questions found in a partially unedited manuscript conserved in Vienna (Cod. Vindob. Pal. 5460).11 Most parts of the text have been inserted in the Questions on the Sentences, but the manuscript also contains some interesting developments devoted to indivisibles, as

6 In Sent. q. 3, pp. 206–268: “Utrum viator intelligat cum continuo et tempore.” 7 In Sent. q. 1, p. 111. 8 In Sent. q. 4, pp. 289–293. 9 In Sent. q. 15, pp. 426–441: “Utrum aliqua res omnino indivisibilis possit esse

longa, lata et profunda.”10 In Sent. q. 16, pp. 442–459: “Utrum tempus sit aliquid positivum reale vel aliqua

res producta a deo vel ab aliquo.”11 For a brief description of the manuscript, cf. Richter, “Handschriftliches zu

Crathorn.”


another version of the question on the continuum12 and some discus-sions on the infi nite.13

In this short paper, we’ll limit ourselves to the edited Questions on the Sentences, without ignoring the aforementioned relevant passages. To begin with, we must examine the mereological principles on which the atomist edifi ce is built.

2. The Mereological Composition of the Continuum

The gnoseological excuse for discussing the divisibility of continuous quantities helps Crathorn to reveal a fi rst apparent paradox in Aristotle’s analysis. If a continuum were composed of an infi nite number of parts, shouldn’t we accept that the wayfarer’s intellect should understand the infi nite when thinking of a continuous body? If not, could we still say, as Aristotle himself asserts in the De memoria et reminiscentia,14 that the intellect knows cum continuo et tempore?15 Though it may seem sophis-tical, the formulation of the problem chosen in this context is not totally innocent and already indicates his fi nitist presuppositions (that a continuum is composed of a fi nite number of parts) and his reduc-tive mereology (that a continuum is nothing but its parts). It is also noteworthy that when these two claims are combined, parts cannot be just potential ones, but rather are actual components of the whole as such. This is the reason why Crathorn presupposes that the wayfarer’s intellect could know (de jure) all the actual parts of a continuous being.

12 In Sent. q. 8: “Utrum continuum componatur ex indivisibilibus, id est ex punc-tis.” This question has been partially edited in Wood, Adam de Wodeham, Tractatus de indivisibilibus, pp. 309–317.

13 In particular q. 9 on the nature of instants (ff. 39va–40rb: “Utrum instans secun-dum substantiam maneat idem toto tempore”) and two others implying the notion of infi nite (q. 22 ff. 55vb–57va: “Utrum ex infi nitate motus extensiva possit concludi infi nitas virtutis intensiva in primo motore”; q. 23 ff. 57va–59ra: “Utrum causa prima possit producere extra se aliquem effectum actu infi nito”).

14 Aristotle, De memoria et reminiscentia, 1, 450a8–9.15 In Sent. q. 3, p. 206: “Utrum viator intelligat cum continuo et tempore. Quod

non videtur. Nullum continuum est intelligibile a viatore. Igitur viator non intelligit cum continuo et tempore. Consequentia patet. Probatio antecedentis, quia si aliquod continuum posset intelligi a viatore, sit illud a; aut est fi nitum aut infi nitum. Si infi nitum, non potest intelligi. Si fi nitum, cum continuum non sit aliud quam partes, sequitir quod viator intelligens a intelligeret omnes partes eius et per consequens intelligeret infi nita, quia omne continuum divisibile est in infi nitum, igitur continet in se infi nitas partes. Ad oppositum est Philosophus in primo libro De memoria et reminiscentia, ubi dicit quod intelligimus cum continuo et tempore.”


This leads Crathorn to a distinctive mereology, according to which a whole is nothing more than the sum of its parts.

The main argument supporting this assertion is the following: if, from an ontological point of view, a whole were different from its parts, one should consider it as a distinct entity; but two absolute things can exist separately without any contradiction, at least thanks to God’s absolute power. It would follow from these premises that God could conserve the whole as existing while destroying its parts, which seems self-contradic-tory.16 It looks rather as if the whole had no ontological existence, for a whole must be identifi ed with the mere continuous addition of its parts.17 This is therefore the fi rst principle of Crathorn’s mereology:

P1: a whole is nothing but the sum of its parts

Philosophers have traditionally opposed to this principle the aporia of diachronic identity. Indeed, shouldn’t we conclude from P1 that a whole is not the same after losing or gaining one of its parts? Socrates’s essence would have changed with one of his hairs being torn off. Anticipating this kind of objection, Crathorn envisaged the existence of essential parts.18 Taking essential parts into account, one should be able to distinguish the identity of a whole W in a given time t, and the identity of a whole W through time, i.e. through a series of instants

16 In Sent. q. 3, p. 206: “. . . totum est suae partes. Hoc probo sic: si totum non est suae partes et totum est aliquid, igitur totum ponit in numerum cum partibus. Con-sequentiam probo, quia sit a multitudo partium, sit b totum, tunc sic: b non est a nec aliqua pars ipsius a. Igitur si b est vera res, b ponit in numerum cum a et cum qualibet parte illius a. Igitur per potentiam dei b totum posset esse non existente multitudine aliqua nec aliqua pars illius, quod falsum est.” As usual in the fourteenth century, God’s absolute power is considered as a logical principle in such a context, because God’s omnipotence is only limited by the principle of non-contradiction.

17 In Sent. q. 3, p. 217: “Propter praedicta videtur mihi quod totum non est alia res ab omnibus suis partibus sibi invicem continuatis.”

18 Crathorn often uses the notion of partes essentiales and though he strongly criticized the Aristotelian categories—notably the very idea of substance—he nonetheless seems to maintain the general distinction between essence and accident. Cf. In Sent. q. 13, pp. 386–402. For a use of the notion of essential part in the continuum debate, see In Sent. q. 3, p. 207 et 223. Essential parts are sometimes called naturae coextensae. Cf. In Sent. q. 13, pp. 386–387: “Istud nomen ‘substantia’ derivatur ab isto verbo ‘substo’ ‘substas’; unde illud proprie vocatur ‘substantia’ quod stat sub alio vel aliis; sed nihil est in isto ligno de quo proprie possit dici quod stet sub aliquo alio, quod est in ligno. Licet enim in ista re sunt multae naturae coextensae, tamen una illarum non est magis sub alia quam econverso. Igitur nulla illarum potest proprie dici substantia.” (italics mine). This precisely means that some parts are essential, but also that all parts have an essence as we shall see.


t, . . ., tn. In a given time t, the identity of W is nothing but the totality of its parts, as an instantaneous ontological photograph, but the iden-tity of W through time is the subset of its essential parts only. But this does not evacuate P1, Crathorn contends, because even at the level of essential parts, one cannot conceive the whole formed by the essence as distinct from its parts.19 Then, even if one wanted to distinguish the essence of a thing from its accidental parts, it should be recognized that P1 works in the same way for the whole set of essential parts. What is at stake here is the generalization of the mereological principle P1, which is applicable to every kind of continua, and not only to natural substances and artefacts. As we shall see, it can also be applied to time and space and in general to all continuous quantities.

Among the several arguments offered by Crathorn,20 one must be carefully examined, because it contains the keystone of the theory, namely, that the composition of spaces is parallel to the composition of things existing in space.

3. The Mereological Composition of Space

To convince his readers, our Oxford master invites them to think about P1 within the category of place. How could we conceive of the place of a whole and its parts if not in a mereological fashion? When a body is divided into its parts, the total place of the whole body is also divided into proportional parts of space. For example, half of a body occupies half of the place of the whole body, and so on if we could divide it again several times. As a corollary of that point, one has to consider that the division of places is strictly parallel to the division of the things existing in those places.

Further, it might be noted that this new element strengthens the argu-ment for P1. This new argument runs as follows: if a whole W were different from its parts, then the place occupied by W must be distinct from the place occupied by its parts—for something always exists in

19 In Sent. q. 3, p. 207: “Deus potest conservare omnes partes istius ligni non destru-endo totum lignum. Sed forte aliquis posset dicere quod quia istae partes sunt de essentia istius ligni, ideo non potest destruere istius ligni, ideo non potest destruere partes istius ligni, nisi destruat totum lignum. Contra: eadem ratione non potest conservare partes istius ligni, nisi conservaret totum lignum; sed partes ligni sunt de essentia ligni.” Cra-thorn declines this argument in many different versions. Cf. p. 207.

20 There are twenty-eight different arguments.


a place; but this sounds nonsensical, otherwise my house could be in London and its parts in Paris.21 Likewise, if we were to divide W in actu, each part would occupy a distinct place and once gathered they would occupy again the whole space of W.

This is an important theoretical move from the composition of a continuum to the composition of space.22 Crathorn thus holds a second mereological principle:

P2: the space occupied by a whole thing is composed of parts of space corresponding to the thing’s parts

This is an important shift that will serve as the basis of Crathorn’s attacks against the divisibilists.23 Indeed, if it is conceivable that a thing is infi nitely divisible, it is far more diffi cult to understand an infi nite division of space. Without determining the number of parts in a con-tinuum for the time being, Crathorn assumes that P1 and P2 imply a third principle concerning the proportional parts in division.

21 Ibid., p. 212: “Si continuum non est suae partes sed res distincta, implicaret con-tradictionem unum corpus esse in loco, nisi plura et distincta corpora essent in eodem loco. Et hoc probo sic: in eodem loco est corpus et suae duae medietates, sicut hoc lignum et suae medietates, et hoc loquendo de toto loco totius ligni. Sed totum lignum est aliud a suis medietatibus. Igitur in eadem loco sunt distincta corpora occupantia totaliter locum illum, scilicet hoc corpus et suae medietates, quia certum est quod totum lignum occupat suum totum locum et duae medietates occupant eundem locum totum. Igitur impossibile est quod aliquod totum occuparet totaliter totum locum, nisi aliqua alia occuparent totaliter eundem locum. Dicitur quod duae medietates non occupant totum locum, sed partes loci, scilicet duas medietates totius, quia sicut totum locatum est alia res a suis medietatibus, sic totus locus non est suae medietates. Et ideo licet duae medietates totius occupent suas medietates loci, non potest concedi ex hoc quod occupant totum locum, sed illud quod occupat totum locum est totum componitur ex duabus medietatibus locati, et non sunt duae medietates . . . Igitur totus locus non est aliud quam suae duae medietates, et per consequens omne corpus locatum est suae duae medietates. Igitur eadem ratione omne totum est suae partes.”

22 We use indifferently ‘space’ and ‘place’, because as we shall see later, Crathorn defi nes the place of a thing as the space it occupies. Cf. In I Sent. q. 14, p. 417. We will turn to this point in the forthcoming sections.

23 Here, it may be suggested that the main source of this theory is the Liber sex principiorum, which is often mentioned by Crathorn, and where a similar approach to space can be found. Cf. Liber sex principiorum [Minio-Paluello] pp. 46–47: “Ubi autem aliud quidem simplex aliud vero compositum; simplex quidem est quod a simplici loco procedit, compositum autem quod ex coniuncto.”


4. The Proportions in Division

According to Crathorn, it follows from P1 and P2 that there should be the same number of parts in the two halves of a thing, and that there shouldn’t be as many parts in a quarter than in the rest of that very thing for example.24 Generally speaking, the number of proportional parts—be it fi nite or infi nite—is conserved through division. It also works out when comparing two different continuous magnitudes: if a piece of wood is twice as big as another, the former has double the number of parts of the latter.25 The third principle is therefore the following:

P3: if a whole W is naturally composed of n parts, if we divide W in x parts, then each of the x parts will be composed of n/x parts

Considered separately, these three principles do not support a particular version of atomism, for the number n of natural parts could still be fi nite or infi nite. Furthermore, Crathorn has not yet given an answer to Aristotle’s anti-atomist critiques. At least, we may assume that if there were an infi nite number of indivisibles in a continuum, one should accept according to P3 that there are unequal infi nities.26 Implicitly, Crathorn seems to reject this claim, because he uses P3 in the series of arguments tending to prove that there is a fi nite number of indi-visibles.27 Indeed, he seems to consider that parts should be countable in some way (at least by God, it might be said). But this is not enough

24 For example, In Sent. q. 3, p. 225: “Secunda conclusio est quod non sunt tot partes in medietate continui quot in toto continuo. Primo quia totum continuum est duplum respectu suae medietatis, sed totum continuum est multitudo omnium partium continui et medietas totus est multitudo partium medietatis, igitur multitudo partium totius continui est dupla respectu multitudinis partium suae medietatis . . .” and p. 226: “Quarta conclusio est quod multitudo omnium partium medietatis continui est aequalis multitudini omnium partium alterius medietatis.”

25 Ibid. p. 226: “Quinta conclusio quod generaliter qualis est proportio continui ad continuum, talis est proportio multitudinis partium unius continui ad alterius. Unde si unum continuum sit duplum ad aliud, multitudo partium illius continui est dupla ad multitudinem partium alterius et sic de aliis proportionibus.”

26 Henry of Harclay, for example, accepts such a possibility. Cf. Murdoch, “Henry of Harclay and the Infi nite.” For the medieval debates on infi nity, see Côté, L’infi nité divine dans la théologie médiévale (1220–1255) and Biard & Celeyrette, De la théologie aux mathématiques. L’infi ni au XIV e siècle.

27 In Sent. q. 3. pp. 226–227.


to prove the fi nite constitution of the continuum. Hence, in support of his fi nitist view, Crathorn adds another stone to the edifi ce:

Everything which is a part of a continuum is either something actual (actualiter aliquid ) or not. One cannot say rationally that it is not some-thing actual, because one cannot understand that what is an actual part of a continuum and belongs to its essence is not something actual or a certain thing. Therefore one must say that a part of a continuum is something actual.28

Indeed, how could a thing be composed of non-things? There are no potential parts in a continuum, but only actual ones, according to P1, because parts constitute the whole as its essence. This fourth principle could be summed up as follows:

P4: the parts of a whole W are all actual parts

The terms ‘composition’ and ‘part’ should thus be taken in a strong sense. The parts in question are the components of reality and can, in principle, exist independently:

But the same things that are called different [pieces of ] wood when they are discontinuous, these very same are called one wood and one whole when continuously joined together, in such a way that the expressions ‘the whole wood’ and ‘one wood’ signify nothing more than the essence of the parts, except from the continuation of these things with each other.29

If we add the compositionality of spaces (P2) to that claim, each part of a continuum is an actually existent thing occupying a single place. Indivisibles are parts of bodies ( partes indivisibiles) and not only mathematical points.30 Therefore, as we shall see in much more detail further down, parts must be conceived in a very strong sense as actual

28 Ibid. p. 227: “Omne id quod est pars continui, vel est actualiter aliquid vel non. Non potest dici rationabiliter quod non sit actualiter aliquid, quia istud non est intel-ligibile quod id quod est actu pars continui et de essentia continui, non sit actualiter aliquid nec res aliqua. Igitur oportet dicere quod pars continui sit actualiter aliquid.”

29 Ibid. p. 217: “Sed illae eaedem res numero, quae dicuntur plura ligna, quando sunt discontinuata, illae eaedem dicuntur unum lignum et unum totum quando sibi invicem continuantur, ita quod isti termini ‘totum lignum’ vel ‘unum lignum’ nihil aliud signifi cant ultra essentiam partium nisi continuationem illarum rerum ad invicem.”

30 Ibid., p. 237: “Septimo conclusio est quod indivisibile est pars corporis con-tinui . . . Nullum corpus continuum fi nitum est divisibile in infi nitum; igitur cuiuslibet corporis continui fi niti est aliqua pars indivisibilis.”


things, with a certain quantity equal to the part of space in which they are located.

On this point, Crathorn parts from his colleague Walter Chatton, who strongly rejects the existence of indivisibles in actu. Chatton’s opin-ion is particularly well expressed in the Reportatio of his Commentary on the Sentences, where he criticizes a point of view similar to Crathorn’s.31 According to Chatton, a continuum can’t be composed of actual indivisibles, for if it is the case, then the whole wouldn’t be a totum per se unum but a mere aggregate of indivisible parts.32 In Chatton’s view, indivisible parts exist are only potentially in the whole, but they would be actual if really divided.

On the contrary, the existence of actual indivisible things in the continuum is one of the main arguments of Crathorn’s fi nitist theory, for one cannot understand an infi nity of actual parts in a fi nite con-tinuum.

5. The Number of Indivisibles

From the elements previously posited, Crathorn infers his critique of Aristotle’s view about the infi nite divisibility of a continuum, as well as Henry of Harclay’s atomist version of it.33 Crathorn clearly asserts that “no fi nite continuum can have an infi nite number of proportional parts”.34 Considering P1 and P4, it follows that if a fi nite continuum were composed of an infi nite number of parts, it would be a self-con-tradictory claim, and the continuum should be actually infi nite, for its

31 Walter Chatton, Reportatio super Sententias [Etzkorn e.a.] II, d. 2, q. 3, pp. 22–23: “Modo volo ego declinare ad hoc quod continuum componatur ex indivisibilibus in potentia, non ex indivisibilibus in actu.”

32 Ibid. p. 126: “Dico quod continuum componi ex talibus in actu includat contra-dictionem, quia eo ipso quod continuum et contiguum differunt oportet quod partes continui uniantur et faciant per se unum, quod si non, non facerent continuum sed contigua esse tantum.” Chatton ascribed the position he is challenging to Democritus (cf. ibid. p. 125).

33 Harclay accepts the existence of indivisibles, but they are infi nite in a continuum. Crathorn frequently refers to Harclay by name. For example, In I Sent. q. 3, p. 259.

34 In Sent. q. 3, p. 226: “Nulla multitudo fi nita est infi nita, sed si aliqua multitudo par-tium alicuius continui esset infi nita, aliqua multitudo infi nita esset fi nita vel aliqua mul-titudine fi nita esset infi nita; igitur nullius continui multitudo partium est infi nita . . .”


parts are always actual according to P4.35 Of course, no Aristotelian philosopher would accept the existence of such an actual infi nite.36

In attempting to explain this argument, Crathorn recalls that follow-ing P1 and P2, “there are as many parts of extension as [parts] of the extended thing”,37 which means that if a continuum were composed of an infi nite number of parts, it would not only be composed of an infi nity of places, but it would also consist in an infi nite extension.38 As a corollary to this fi rst philosophical consequence, it should be noticed that to accept the infi nite division of places would imply the idea of an infi nite number of places inside the continuum according to Crathorn, i.e. the existence of an infi nite place by composition and addition.39 Moreover, one must also admit the possible existence of an infi nite body, for “if there were an infi nite number of proportional parts in a fi nite continuum, this fi nite body according to the dimension of place would stretch to the infi nite”.40 Admittedly, such an infi nite body can’t exist according to the Aristotelian cosmology.41 In such a context, it is clear that P2 and the notion of locus are the most important require-ments for Crathorn’s position.

The aforementioned strategy consists in showing that from a mereo-topological point of view, the only conceivable infi nity of parts is actual

35 Ibid.: “Si igitur in quolibet continuo sint actualiter infi nitae partes, quod oportet dicere si in quolibet continuo sint infi nitae partes, sequitur quod quodlibet continuum est actualiter infi nitae partes. Igitur eadem partes continui sunt actualiter fi nitae, quia sunt continuum fi nitum, et sunt actualiter infi nitae, quia in quolibet continuo sunt infi nitae partes . . . igitur eadem partes sunt fi nitae et non fi nitae.”

36 Aristotle’s critique of the possibility of an actual infi nity is well-known and it is not necessary to review it in this context. Cf. Physics, book III and VI in particular. For the medieval background, see Murdoch, “Infi nity and Continuity.”

37 In Sent. q. 3, p. 227: “Tot sunt partes extensionis quot rei extensae.”38 Ibid. p. 227: “. . . sicut tota res extensa correspondet toti extensioni, sic medietas rei

extensae medietati extensionis, et secunda pars proportionalis rei extensae correspondet secundae parti proportionali extensionis, et sic de aliis partibus proportionalibus. Igitur si continuum est infi nitae partes, est infi nitae partes extensae. Igitur si continuum fi nitum est infi nita secundum multitudinem, ista est infi nita secundum extensionem.”

39 Ibid. p. 228: “Aliter potest dici quod partes continui sunt infi nitae secundum multi-tudinem sed fi nitae secundum locum, quia sunt in loco fi nito. Sed contra: locus non est aliud quam partes loci; sed partes loci sunt infi nitae secundum se si partes rei locatae sint infi nitae, quia medietas totius locati est in medietate totius loci et medietas medietatis locati est in mediate medietatis loci; igitur quot sunt partes proportionales illius locati tot sunt partes proportionales illius loci. Sed multitudo partium proportionalium illius locati est infi nita; igitur multitudo partium proportionalium illius loci est infi nita.”

40 Ibid. p. 229: “Si in aliquo continuo fi nito essent partes proportionales infi nitae, illud corpus fi nitum secundum omnem dimensionem loci extenderetur in infi nitum.”

41 See for example, Aristotle, De caelo, I, 5 et 7.


and not just potential, because if we start thinking from the point of view of the part-whole relation we cannot consider parts as mere potential units. Here, the most original feature of the argument rests on the use Crathorn makes of the notion of place. Indeed, beyond his adherence to the four principles P1 to P4, it follows that to each indivisible part corresponds a single place and a single position (situs punctualis), and therefore a kind of quantity, as we will see in the next chapters. If not, the arguments of the infi nite extension and the infi nite body would fail.

It is illuminating to compare Crathorn’s solution to similar argu-ments given by some of his contemporaries. As an example, Gerard of Odo in Paris, who wrote his Commentary on the Sentences and other small tracts on the continuum a few years earlier than Crathorn, also uses the argument of actual infi nities for his own fi nitist theory.42 Its form is indeed quite the same: if we suppose that a continuum is composed of an infi nite number of parts, this continuum should be itself actually infi nite.43 Odo’s followers and critics will also report this argument as a major one. For example, it is to be found in John the Canon’s Questions on the eight books of the Physics 44 and in Gaetano of Thiene’s Collection on the eight books of the Physics.45 Even infi nitists such as Nicholas of Autre-court and Nicholas Bonetus used this argument.46

Whether or not Crathorn is the fi rst to have used this argument, he nonetheless has a particular place in this story, for it isn’t likely that he was infl uenced by Gerard of Odo or John the Canon, who both were

42 See Sander de Boer’s contribution in this volume for the context of the argument.43 Gerard of Odo, De continuo, Ms. Madrid, Bibl. nac. 4229, ff. 179rb–va (quoted by

S. de Boer in this volume): “Omne totum compositum ex magnitudinibus multitudine infi nitis, sicut componitur cubitus ex duobus semicubitis, est magnitudo actu infi nita. Sed non est dare magnitudinem actu infi nitam. Ergo nullum continuum est divisibile in infi nitum.”

44 John the Canon, Quaestiones super octo libros Physicorum Aristotelis [Venice, 1520], f. 59vb: “De totum compositum ex partibus vel magnitudinibus multitudine infi nitis, sicut componitur cubitus ex duobus semicubitus, est magnitudo infi nita; sed nullum continuum est magnitudo actu infi nita.”

45 Gaetano of Thiene, Recollectae super octo libros Physicorum [Venise, 1496], f. 38vb: “. . . si continuum componitur ex semper divisibilibus ipsum in infi nitum excedit aliam magnitudinem et est actu infi nitum . . .”

46 For example, Nicholas of Autrecourt, Exigit ordo [O’Donnell], p. 212, ll. 29–42: “Dicerent autem contra hoc forsan: si punctum additum puncto faciunt majus et extensionem quamdam et tres faciunt majus quam duo et sic semper et ibi sint infi nita, sequitur quod ibi sit infi nita extensio.” On this point, cf. Grellard, “Les présupposés méthodologiques de l’atomisme: la théorie du continu de Nicolas d’Autrécourt et Nicolas Bonet.”


in Paris at that time, or by Nicholas of Autrecourt who wrote his Exigit ordo at the very same time. It seems more likely that Crathorn and his contemporaries were inspired by their Oxonian predecessors Henry of Harclay and Walter Chatton.

The Chancellor of the University of Oxford, Henry of Harclay, affi rmed that a continuum is composed of an infi nite number of indi-visibles, but he accepted only one sort of actual infi nite: the straight line.47 But according to him it is not rational to extend this for a sur-face or a body. As a consequence, Harclay considered indivisibles as potential parts and would probably have refused Crathorn’s mereo-logical principles. Nevertheless the attack on actual infi nitism already existed in embryonic form in Harclay’s position, and Crathorn probably found his starting point in it, adding to this intuition his mereological principles.

More probable still is Walter Chatton’s infl uence on Crathorn,48 for he will use a similar argument based on the actual infi nite, although he didn’t attach as much importance to the notions of locus and situs.49 One of the reasons for this difference of opinion is that indivisibles are still considered as potential parts of the whole continuum in Chatton’s view.50 They cannot exist separately, therefore they cannot have a single place on their own.

From this point of view, Crathorn’s position is quite original in the atomist family. While using a common matrix of arguments, he is the only one who emphasizes both the mereological composition of

47 On this point, cf. Murdoch, “Henry of Harclay and the Infi nite” (in particular p. 233).

48 It might be noticed that Crathorn doesn’t agree with Chatton on several points. For references to Chatton, see In Sent. q. 3. pp. 261 and 265.

49 Walter Chatton, Quaestio de continuo, in. Murdoch & Synan, “Two questions on the continum: Walter Chatton(?), O.F.M. and Adam Wodeham, O.F.M.,” p. 258: “. . . responsio ad primum: dico quod sunt fi nita. Ad philosophum dico quod continuum esse divisibile in infi nitum potest <dupliciter> intelligi: vel quod in continuo sint actu infi nite partes, quarum quelibet est extra aliam et nulla est alia, que possunt dividi ad invicem, et hoc est falsum et contra rationem . . .”

50 See the texts quoted above at the end of the previous section. See also Walter Chatton, Quaestio de continuo, in. Murdoch et Synan, “Two questions on the continuum: Walter Chatton(?), O.F.M. and Adam Wodeham, O.F.M.” p. 246: “Istis suppositis, teneo 3 conclusiones. Prima: quod non componitur continuum ex indivisibilibus in actu, quia inter terminos est contradiccio ‘continuum’ et ‘indivisibile in actu’ quia, si sit continuum, igitur partes eius sic se habent quod nulla est in actu per se existens separata ab alia; et si sit indivisibile in actu, est per se existens separatum ab alio eiusdem rei. Oppositum dicit Democritus ponens continuum fi eri ex athomis, tantum per congregationem quandam continuatis . . .”


things and spaces as well as the actuality of parts.51 But his singularity comes from his use of the notion of locus. Indeed, the arguments of the actual infi nity of parts and infi nite extension require P2 and the coincidence between indivisible parts and single places for its effi cacy. Moreover, Crathorn contends that ‘whole’ and ‘part’ are only imposed to signify the local conjunction of indivisible places.52 In this respect, he is probably one of the more consequent atomists, because if an indivisible were not considered as an indivisible part, with a kind of extension resulting from the minimal place it can fi ll, all the previous argumentation would fail. Gerard of Odo, for example, whose theory is very close to Crathorn’s, is not consistent when he seems to consider indivisibles as unextended.53

Here again, Crathorn is indebted to Harclay’s analysis of contact between indivisibles when he tries to defi ne contiguity and continuity of atoms thanks to the central notion of place.

6. Contiguity and Continuity

Aristotle’s well-known paradox formulated in the sixth book of the Physics arises when asking the atomist how two atoms can generate an increase in size.54 It raises the problem of the contact between indivisibles, for two things can touch together parts to parts, whole to whole, or parts to whole, Aristotle said. Having no parts, atoms can

51 Except from Crathorn, there were no other philosophers at Oxford, as far as I know, who held this view about the actuality of indivisible parts of a continuum. In Chatton’s Reportatio, the reportator ascribes this position to some contemporaries. Cf. Walter Chatton, Reportatio super Sententias [Etzkorn e.a.] II, d. 2, q. 3, p. 136: “Sed secundum usum modernorum, potest componi ex actu indivisibilibus, quia vocat actu tale quod est tale extra animam et extra causam, quantumcumque non sit separatum ab alio; componitur ergo ex actu indivisibilibus, id est ex non habentibus partes.” (ital-ics mine) It’s impossible to know to whom Chatton was refering in this text (written around 1322–23). If we believe Gregory of Rimini, the fi nitist position, with actual or potential indivisibles, was common among his contemporaries. Cf. Gregory of Rimini, Lectura super secundum Sententiarum [Trapp], t. II, d. 2, q. 2, p. 278: “Nec potest dici, sicut communis dicunt tenentes huiusmodi compositionem ex indivisibilibus, quod componatur ex fi nitis tantum indivisibilibus.” We don’t know whether Gregory had Chatton, Odo or Crathorn in mind, or some other unknown philosopher. At the end of the fourteenth century, Wyclif held a position very similar to Crathorn’s own thesis. For Wyclif, see the references in the next sections.

52 In Sent. q. 3, p. 224: “. . . hoc nomen ‘totum’ et hoc nomen ‘pars’, . . . imponuntur a locali coniunctione rerum, quae coniunctio non est aliud quam loca vel partes loci.”

53 As it is revealed in the texts quoted by Sander W. de Boer in this volume.54 Aristotle, Physics VI, 1, 231a29–b6.


only touch whole to whole, and this makes no increase in size but a mere superposition; now, superposition non facit maius.55

When replying to this famous objection, Henry of Harclay claims that Aristotle missed the point, because the Stagirite used to consider indivisibles in one and the same position (situs), but according to him, indivisibles can touch according to different positions (secundum distinctos situs) and this way they can cause an increase in size.56 Henry of Haclay doesn’t develop this point further, but Crathorn will explain this point very clearly.57

To my mind, the interesting idea that emerges from these topological elements is the idea of a reference landmark, thanks to which atoms can be located by their respective positions. This is not totally extra-neous to Aristotle’s cosmology, since similar tools can be found in the De caelo for example.58 Moreover, Aristotle sometimes defi nes a point as a substance with a position, as in the Posterior analytics.59 But the appeal to locus and situs of indivisibles is problematic for philosophers

55 On this argument and the consecutive medieval debates, see Murdoch, “Super-position, Congruence and Continuity in the Middle Ages.”

56 Henry de Harclay, Quaestio de infi nito et continuo, Mss Tortosa Catedral 88, f. 89r and Florence, Biblioteca Nazionale, Fondo principale, II. II. 281, f. 98r–v (quoted in Murdoch, “Henry of Harclay and the Infi nite,” p. 244): “Sed, licet hec responsio suf-fi ceret ad hominem, non tamen est realis responsio. Et ideo pono aliam et dico quod indivisibile tangit indivisibile secundum totum, sed potest hoc esse dupliciter: vel totum tangit totum in eodem situ, et tunc est superpositio sicut dicit Commentator, et non faciunt infi nita indivisibilia plus quam unum. . . . Et ideo dico quod non propter indi-visibilitatem quod unum indivisibile sic additum indivisibili non facit maius extensive, sed quia additur ei secundum eundem situm et non secundum distinctum situm. Si tamen indivisibile applicetur immediate ad indivisibile secundum distinctum situm, potest magis facere secundum situm.”

57 Gerard of Odo, for example, will follow Harclay’s text to the letter. Cf. Gerard of Odo, De continuo, Ms. Oxford, Bodleian Can. Misc. 177, f. 230v (quoted in Murdoch, “Superposition, Congruence and Continuity . . .,” p. 435): “Dico quod totum indivisi-bile tangit totum aliud indivisibile, non tamen secundum omnem differentiam situs, sed secundum unam tantum, scilicet secundum ante vel secundum retro et sic de aliis. Unde si unum indivisibile tangeret aliud indivisibile secundum omnem differentiam loci, scilicet secundum ante et retro et secundum alias differentias omnes, tunc bene sequitur quod indivisibilia non essent loco discreta nec constituerent aliquod maius. Sed si unum tangit reliquum secundum unam differentiam loci, ideo sunt loco discreta et constituunt aliquod maius.”

58 Cf. Aristotle, De caelo, II, 2.59 Aristotle, Posterior analytics, I, 27, 87a36. As Rega Wood points out in her chapter

in this volume, this surprising assertion by Aristotle has been ignored by a majority of medieval philosophers, although Robert Grosseteste paid some attention to it in his commentary. Cf. Robert Grosseteste, Commentarius in Posteriorum analyticorum libros, 1, 18 [Rossi], p. 258.


such as Henry of Harclay or Gerard of Odo, who do not conceive of indivisibles as occupying places. In fact, how could indivisibles occupy a place if they are neither extended nor actual things?

Again, Aristotle himself raised this point explicitly. When criticizing the atomists in De generatione et corruptione (I, 2) Aristotle repeated the argument from the Physics saying that it is not possible for a quantity to come from non-quantitative things.60 He immediately added: “Fur-thermore, where will the points be?”61 When he turned back to the problem of the continuum in chapter 6 of book I, Aristotle insisted on the notion of contact, thanks to which continuity can be conceived. But, as he contended, the notion of contact cannot be understood without the notion of position, which in turn relies on the notion of place.

Nevertheless, ‘contact’ in its proper sense belongs only to things which have a ‘position’, and ‘position’ belongs to those things which have also a ‘place’.62

Touch always occurs between two situated things because things in contact need to be in a certain spatial relation. These touching things, Aristotle said, also need to have a certain discrete magnitude, for if not they cannot touch.63 So, when reading the De generatione et corruptione, the medieval philosophers could fi nd the key to the problem of contigu-ity and continuity: to ascribe a place and a certain magnitude to the indivisibles. But other sources could be used, as the Liber sex principiorum which is sometimes cited by Crathorn and where one can fi nd a similar notion of indivisible places for points and body’s minima.64

Due to the need to consider indivisibles as occupying a single place, Crathorn is able to reconstruct contact between things and their con-tinuity from the notions of locus and situs. As he told us in a passage of his Questions on the Sentences:

60 Cf. Aristotle, De generatione et corruptione, I, 2, 316b5.61 Ibid.62 Aristotle, De generatione et corruptione, I, 6, 322b30–323a5.63 Ibid.64 Liber sex principiorum [Minio-Paluello] pp. 46–47: “Locus autem simplex est origo

et constitutio eius quod continuorum est, locus vero (ut dictum est quidem) compositus habet particulas quidem ad eundem terminum copulatas ad quem et corporis parti-cule coniunguntur, corporis vero partes ad punctum. Loci ergo partes iuxta punctum necesse fi eri; erit itaque locus simplex in quo punctum adiacere constabit, loci vero particule soliditatis particulas claudunt; etenim loca quidem simplicia minimi corporis occupativa sunt.”


What continuity in fact is will become clear below when asking whether God is ubiquitous (q. 15). There it will be said, indeed, what is place, and once the nature of place is understood, one can easily see what is continuity and discontinuity, and what is contiguity.65

John E. Murdoch has already noticed that the notion of situs used by Harclay implies a physicalist version of the indivisible, even if Har-clay doesn’t develop his view in this way.66 If Harclay didn’t take the plunge, Crathorn seems to have thought of indivisibles in this physi-calist fashion.

Crathorn’s goal is to redefi ne the problematic notions in Aristotle’s critique of atomism. If two things are continuous, according to Aristotle, their limits have to be one, and if they are contiguous, the extremities of the things must be together.67 The strongest attack against atomists comes from this characterization of continuity and contiguity. Therefore, Crathorn suggests an alternative defi nition:

[contiguity] has to be defi ned in this way: “contiguous things are situated and located things, between which there is no intermediary place or posi-tion,” and this defi nition suits for bodies, surfaces, lines and points.68

It should be objected to this defi nition that contiguity never explains continuity.69 But since Crathorn adds the clause of non-existence of interparticulate spaces between indivisibles, contiguity becomes con-tinuity, because they produce a new thing without any vacuum in it. Therefore, properly speaking, a continuous thing is full of matter, there is no empty space in it, no void. This is true, in slightly different senses, for every kind of entity, including time.

65 In Sent. q. 4, p. 218: “Quid vero sit continuitas patebit infra, cum quaeretur utrum deus sit ubique; tunc enim dicetur quid est locus, et intellecto quid est locus, cito potest videri quid est continuatio et quid dicontinuatio et quid contiguitas.”

66 Murdoch, “Henry of Harclay and the Infi nite,” p. 244: “It is clear that in claim-ing that indivisibles can touch according to distinct positions Harclay was considering these indivisibles not as absolutely extensionless entities they really were, but as if they were physical things.” It is not absolutely evident that Harclay was not aware of this physicalist consequence. Nevertheless, it would be incompatible with his infi nitism. On the contrary, it is clear enough that Crathorn endorsed consciously these physicalist implications.

67 Aristotle, Physics, VI, 321a21–25.68 In Sent. q. 3, p. 255: “Sed debet sic defi niri: ‘contigua sunt situata vel locata, inter

quae non est locus vel situs medius’, et ista defi nitio competit corporibus et superfi -ciebus, lineis et punctis.”

69 We fi nd this argument in Chatton’s Reportatio for example. Cf. Reportatio super Sententias [Etzkorn e.a.], d. 2, q. 3, p. 126.


For a thing to be continuous is nothing but its parts being joined together in a local or a temporal way, without any intermediary place or time [between them], and these parts being joined, they hold together or mutually succeed in place or in time without subtraction of place or time. It follows that the continuity of a body, of a line or a surface, is the continuity of the parts of place, because parts are said to be continuously located according to the continuity of the parts of place . . .70

From the beginning, Crathorn presupposes that an indivisible occupies a single place and that it must consequently have a certain quantity. This is implied by the argument from actual infi nity and also from the aforementioned defi nition of contiguity and continuity. Aristotle himself thought that position and place should imply weight or lightness in some sense.71 Indeed, Aristotle’s target was mainly the possibility of touching unextended points, but not of touching things (i.e. something with a place, a position and a certain extension). Crathorn’s reappraisal of contiguity and continuity seems to work as follows: Aristotle was right in claiming that extensionless points cannot touch, but if points or indivisibles were given a certain place and extension, they could touch according to their contiguous positions. Crathorn’s theory is in some sense a reinterpretation of Aristotle, paying attention to the physical possibilities of his cosmology.

The keystone of Crathorn’s atomist theory is thus to have consid-ered that indivisibles may have a sort of quantity and extension. How are the notions of place, extension and quantity linked in Crathorn’s thought?

7. Extended and Qualified Atoms

In a long discussion on the category of quantity (q. 14) and on the quantity of indivisibles in particular (q. 15), Crathorn incidentally

70 In Sent. q. 16, pp. 456–457: “Rem esse continuam non est aliud quam partes illius rei sibi invicem coniungi localiter vel temporaliter sine loco vel tempore medio et tales partes sic coniunctas simul teneri vel sibi invicem succedere vel loco vel tempore sine interceptione loci vel tempori. Unde continuitas corporis vel lineae vel superfi ciei est continuitas partium loci, quia continuitate partium loci dicuntur partes locatae continue . . .”

71 Aristotle, De generatione et corruptione, I, 6, 323a10: “Now, since position belongs to such things as also have a ‘place’, and the primary differentiation of ‘place’ is ‘above’ and ‘below’ and other such pairs of opposites, all things which are in contact with one another would have ‘weight’ and ‘lightness’, either both of these qualities or one of them.”


examines the nature of place.72 This addition is not fortuitous as we shall see.

The nature of quantity was probably one of the most debated topics in the fourteenth-century natural philosophy. The question usually was to know whether quantity is a distinct and real category or if it is reduc-ible to substance and/or quality for example. As an example, William of Ockham endeavoured to reduce the ten Aristotelian categories to mere names and concepts, but he thought that there really were exist-ing singular substances and singular qualities in the world, which are signifi ed by terms from the ten categories. Substance terms (as ‘homo’) signify only singular substances, quality terms (as ‘albedo’) signify only singular qualities; the other eight categories signify only substances and qualities and are distinguished by the way they refer to them, i.e. by their connotations. On the contrary, other authors such as Walter Burley will accept other categories as real. The medieval positions are so numerous that it is impossible to mention them all here.73

Concerning categories in general, Crathorn holds a strong and radical view, even more than Ockham’s own position, for he thinks that the very same thing can be either a substance, a quality or a quantity.74 It would be too long to detail Crathorn’s analysis of the aristotelian categories in this paper,75 but his aim is to show several points: that there are only atoms and that no one can be called properly a substance with respect to the others; that quantity is nothing real except the situated atoms; that quality also fl ows from the very nature of the atoms; that place is nothing else than the local organization of atoms; etc. On quantity in particular, Crathorn invites us to distinguish between two kinds of quantitas: according to dimension (quantitas dimensiva) and according to perfection or value (quantitas perfectiva).

To begin with the quantitas perfectiva, Crathorn is quick to admit that it is not something different from the thing itself, because the value or perfection of some entity is nothing else than its very nature or essence. Nonetheless, though things have a natural perfection, they are not

72 In Sent. q. 14, p. 411: “Hic interponitur una questio de loco . . . Utrum locus proprie loquendo de loco sit aliquid reale productum a Deo vel a creatura.”

73 For an overview of some of these positions, cf. Biard & Rosier-Catach (eds), La tradition médiévale des catégories; and Lamy, Substance et quantité à la fi n du XIII e et au début du XIV e siècle.

74 In Sent. q. 17, p. 462: “Secunda conclusio est quod eadem res numero est sub-stantia, quantitas, qualitas . . .”

75 I am preparing a paper on this topic.


equivalent to one another, for they can be compared. Two equivalent portions of different essences are not equivalent: a bit of gold is not equivalent to a bit of lead, even if they have the same weight and dimension. And this is perfectly appropriate for indivisibles as well. An indivisible is an actual thing, it has a natural perfection, which can be compared with others: an atom of gold is not equivalent to an atom of lead.76

As a consequence, indivisibles are not the undifferentiated atoms of Democritus or Epicurus. There are atoms of gold, of lead, etc. This sounds as a modern chemical theory of elemental atoms, but Crathorn prefers to call ‘points’ or ‘indivisibles’ the minimal parts of things rather than ‘atoms’. After accepting that there are points of gold, could he admit that those indivisibles have any quantity, in the sense of an extension or a dimension? Once again, Crathorn still presup-poses that indivisibles have a kind of extension when explaining what he understands by ‘points of gold’, but he never explicitly says that they are extended.77

Turning then to the quantitas dimensiva, Crathorn refuses, as for quan-titas perfectiva, to admit that this kind of quantity should be something different from the thing itself. Quantity does not have an independent mode of being. But Crathorn also refuses to identify dimension with the thing itself. Of course, quantity depends on the indivisibles and their order, but cannot be identifi ed with them, because a thing can increase or decrease in size without changing its nature.78 A compressed sponge remains the same sponge in essence. Then, how can dimension be apprehended? Crathorn’s answer is not surprising: quantity according

76 In Sent. q. 14, p. 405: “Secunda conclusio est quod res omnino indivisibilis, quae scilicet non habet partem extra partem nec partem inexistentem parti, est quanta secundum perfectionem et valorem, cuiusmodi sunt deus et angeli, sicut communiter ponitur. Similiter talis res est punctum auri omnino indivisibilis, quia scilicet non habet partem extra partem nec partem inexistentem parti, et hoc patet sic: Talis res essentialiter excedit aliam et aequivalet plures res sicut punctum auri praedicto modo indivisibile essentialiter excedit punctum plumbi consimiliter indivisibile et aequivalet plura puncta plumbea praedicto modo indivisibilia. Igitur tales res indivisibiles sunt essentialiter quantae et quantitates illo modo, quo res dicuntur quantae secundum perfectionem et valorem.” (italics mine).

77 In Sent. q. 14, p. 405: “Quod autem sint talia puncta aurea, quae sunt partes auro omnino indivisibilis, quae puncta non habent partem extra partem nec partem inexistentem parti, probatio: quia sit a unum punctum aureum habens partem extra partem, quod opportet concedere de quolibet auro fi nito scilicet quod habeat partem sui, que pars non habet partem extra partem; aliter aurum fi nitae longitudinis esset infi nitae longitudinis, sicut probavi supra . . .”

78 Cf. In Sent. q. 14, pp. 406–407.


to dimension is nothing other than the place or space occupied by the quantifi ed thing.79 Here we are, the notion of locus is back.

Now, how to defi ne place? First of all, place is immobile, it is not a real thing and it is not the limits of a body.80 What does it mean when it is said that place is not a real thing produced by God? It means that place is not a real category, no more than quantity. When a part of space is not occupied by something, it is empty, that is to say it is void, which means that place is fi rstly a purum nihil.81 Therefore, the classical arguments of impenetrability of bodies and place don’t work here, because space itself is nothing real. Thus, the same space can be called either void or place, and even quantity or dimension. Place is a part of space (according to P2) fi lled by something having a certain quantity and a certain position. It functions as a system of measure or as a natural cadastre with no ontological reality, and it simply allows us to determine the position of each indivisibles. The quantitas dimensiva of a thing is thus defi ned by the space it occupies, i.e. by its place in the

79 In Sent. q. 14, p. 411: “. . . quantitas dimensiva vel dimensio rei dimensionatae est dimensio spatii, in quo est res, et partes dimensionis sunt partes spatii, ita quod longitudo aeris non est aliud quam longitudo spatii, in quo aer est, et latitudo aeris est eiusdem spatii latitudo, et profunditas eiusdem spatii profunditas. Et idem est intel-ligendum de longitudine, latitudine et profunditate cuiuscumque rei longae, latae et profundae, et istud patet quasi ad sensum, quia rem esse longam nihil aliud est quam rem esse in longo spatio . . .”

80 In Sent. q. 14, p. 412–413: “Ad cuius intellectum primo probo istam conclusionem quod de ratione loci est quod sit omnino immobilis . . . Secunda conclusio est quod locus non est aliquid positivum reale . . . Tertia conclusio est quod locus non est ultimum corporis continentis loquendo proprie de loco.”

81 In Sent. q. 14, p. 417: “Ad quartum dicendum quod quando nihil est in spatio, id est, quando nulla res positiva est in spatio, tunc spatium non est locus sed vacuum; quando autem aliquid est in spatio vel ponitur de novo in spatio, tunc id idem, quod prius fuit vacuum et non locus, fi t plenum et locus. Vacuum esse non est impossibile, sed necessarium, quia extra caelum est vacuum infi nitum, in istis autem inferioribus per potentiam Dei posset esse vacuum et est pro aliquo tempore . . . Ad sextum dicendum quod locus nihil est, et concedo quod locatum est in illo quod est pure nihil, et potest poni in illo quod est pure nihil, sicut Deus posset ponere hominem in spatium extra caelum, et tamen illud spatium est pure nihil”. Aristotle already pointed out that, accord-ing to his adversaries, the notion of void implies the notion of a place deprived of a body. Cf. Physics IV, 1, 208b26. Cf. Grant, Much Ado about Nothing, pp. 9–13. Elsewhere, Crathorn defi nes ‘place’ as an imaginary or purely intellectual thing: “Locus vero non est aliqua res accipiendo hoc nomen ‘res’ primo modo, sed tantum est res intelligibiliter et imaginative proprie loquendo de loco, ut infra ostendetur.” (q. 3, p. 224).


general world’s landmark.82 It is worth noticing that this conception of space is fairly original in the Middle Ages.83

On several occasions, Crathorn seems to admit that an indivisible can occupy a punctual place, for he always uses P2 and its fi nitist corollary in this context,84 and accepts the existence of locus punctualis.85 Unfortu-nately, it is never specifi ed whether punctual place is either one, two, or three-dimensional. But if punctual place were one or two-dimensional, this wouldn’t help the demonstration of the existence of indivisibles, for the aporia of touching points would still be threatening with regard to places. How could unextended places touch each other? Anyway, from P2 and from the conclusion that the number of indivisibles is fi nite, we may assume that punctual places can be at least in some cases three-dimensional—in bodies for example. In q. 15, even if he is more concerned with the cases of angels and souls, Crathorn concludes:

To the question I say that it doesn’t seem to me irrational to say that something absolutely indivisible be essentially long, large and deep, if something absolutely indivisible could be essentially in a long, large and deep space.86

It is important to note that this does not mean that indivisible parts and places are extended in the same way as a body, because indivisibles have no confi guration in space. They are qualifi ed, extended in some way, but positions cannot be distinguished inside an indivisible, that is to say that points do not have sides, orientation, nor angles. Nevertheless, as a locus punctualis is contiguous to another and both cannot coincide,

82 In Sent. q. 14, p. 419: “Locus et locatus sunt aequalia secundum dimensionem . . . et isto modo res locata dicitur aequalis suo loco vel spatio in quo est, quia locus vel spa-tium est dimensio vel dimensiones ipsius rei locatae.”

83 For an overview of medieval doctrine of space, see Grant, “Place and Space in Medieval Physical Thought” and idem “The medieval Doctrine of Place: Some Fun-damental Problems and Solutions”. Let us note that Crathorn’s description of space seems far closer to the one professed by Francesco Patrizi (1529–97) than to the views of his contemporaries.

84 In Sent. q. 14, p. 419: “. . . res locata dicitur aequalis suo loco vel spatio, in quo est, quia ipse locus vel spatium est dimensio vel dimensiones ipsius rei locatae. Isto habito arguo sic: quanta est multitudo partium spatii vel loci, tanta est praecise multitudo partium rei locatae situ et loco distinctarum, neque maior neque minor.”

85 For example, In Sent. q. 16, p. 456. We will turn to this notion with more details in a subsequent section when examining the nature of motion (section 8).

86 In Sent. q. 15, p. 440: “Ad quaestionem dico quod non apparet mihi irrationabile dicere quod aliqua res omnino indivisibilis essentialiter sit longa, lata et profunda, si aliqua res omnino indivisibilis essentialiter possit esse in spatio longo, lato et profundo.”


three points or three indivisible places may form a continuum and each part will be situated relatively to the other with a certain orientation, and with angles.87

Therefore, what he calls ‘points’ should have a certain quantity or dimension, even though they are indivisibles. Crathorn’s originality is thus to have developed some intuitions concerning the importance of the notion of place already present in some other philosophical texts.88

Now that the nature of indivisibles has been reconstructed, the last points still to be dealt with are the principles of Crathorn’s atomist physics we announced at the beginning of this paper. How could one explain natural phenomena from this minimal analysis of the ultimate components of reality? Let us turn fi rst to the explanation of motion, from which the rest will follow.

8. The Explanation of Motion

As a consequence of the continuum’s mereotopological structure, motion will be defi ned as a local motion of atoms. Just as the continuum is composed out of indivisibles, motion is made of atoms of motion.89 If we analyse in detail the nature of motion, it is composed of parts which are atoms covering a certain space in a determinate time. In other words, Crathorn says:

What is moved was in a certain place, in which it is not for the time being, and will be in a certain place, in which it is not for the time being,

87 In Sent. q. 3, p. 259: “. . . licet nihil sit rectum vel obliquum nisi divisibile et ideo punctum non est rectum nec obliquum, quia est indivisibile, tamen unum punctum potest continuari alteri puncto secundum situm rectum . . .”

88 Chatton however was close to discovering a similar solution when he wrote: “Ad secundum: quid vocas esse quantum? Si quod habeat partes eiusdem racionis, dico quod indivisibile non habet partes, nec est quantum, quia includit contradiccionem; si quod sit pars quanti, vel quod sit talis res que cum alia re eiusdem racionis componit quantum, concedo.” (Quaestio de continuo [Murdoch & Synan], p. 259). Nicholas of Autrecourt will also have this kind of intuition in the Exigit ordo [O’Donnell], pp. 207–208: “. . . vel per quantum intelligis habens propriam situalitatem et esse circumscriptive et sic concedo quod punctum potest dici quantum, et tunc non poteris secundum hoc concludere ex non quanto fi eri quantum.” See also the brief description of Wyclif ’s atomism in the section 10 of this chapter. For more details on Wyclif, see Emily Michael’s chapter in this volume.

89 In Sent. q. 16, p. 443: “Circa primum sciendum quod sicut supra tenui quod corpus continuum componitur ex indivisibilibus, ita teneo quod motus componitur ex mutationibus subitis.” Here Crathorn doesn’t use the vocabulary of mutata esse, but the idea is quite similar.


speaking of places of this space which are covered by the mobile through this motion.90

Of course, this general defi nition of motion is suitable for bodies as well as for indivisibles.91 As indivisibles can be contiguously situated, all that is necessary for them to be moved is to pass from a punctual place to another contiguous punctual place in a given time made of a certain number of instants. This conception of motion requires the existence of void, for an atom must arrive in an empty punctual place when moving, and we have seen that Crathorn accepts this point with-out any kind of hesitation.

How to understand the continuity of motion from this general account of local motion? Here, one must always have in mind the idea of a cadastral system. As Crathorn wrote, “one must know that time is to motion what place is to body”.92 The parallelism of space and time goes much further, because as place is nothing really existing but only empty space that could be fi lled out by an atom or a body, time and instants too are nothing but a measure of length, they are not things (res).93 “As place is the measure of a continuous body, time is [the measure] of a continuous motion”.94 Each point occupies a distinct position in space and can change from one place to another in one or more instants of time. Therefore, a continuous motion will be the one by which a mobile passes from one place to another contiguous one in one instant—i.e. passing from an instant to another one contiguous to the former. But this theory of motion raises some new problems, in particular for the understanding of variations of speed.

90 In Sent. q. 3, p. 255: “. . . illud quod movetur fuit in aliquo loco, in quo modo non est, et erit in aliquo loco, in quo modo non est, et loquendo de locis illius spatii quod mobile pertransit illo motu.”

91 In Sent. q. 3, p. 255: “Ad quartum dico quod punctum potest moveri et movetur quandocumque corpus movetur . . .”

92 In Sent. q. 16, p. 455: “Sciendum igitur quod tempus se habet ad motum sicut locus ad corpus.”

93 In Sent. q. 16, p. 450: “Instans vero se habet ad mutationem sicut locus punctualis ad punctum. Sicut enim locus punctualis non est punctum vocando punctum aliquid reale indivisibile situaliter nec est aliquid reale positivum distinctum realiter a puncto, sed spatium indivisibile, in quo est tale punctum reale positivum, sic instans non est mutatio vocando mutationem aliquid positivum reale subito et indistincte acquisitum alicui nec est aliquid positivum reale distinctum ab illo, sed est duratio indivisibilis, in qua duratio talis acquiritur.”

94 In Sent. q. 3, p. 242: “Sicut locus est mensura corporis continui, sic tempus motus continui.”


9. The Analysis of Speed

An indivisible cannot pass from a punctual space to another one which is not contiguous without passing through all the punctual spaces in between. It cannot jump from one place to another distant one.95 Variations of speed can only be explained by the variation of time a mobile can take to move from one place to another. This rather puz-zling assertion entails another strange conclusion: given the fact that punctual spaces and instants are the indivisible units for measuring motion, if a mobile goes from one punctual space to the next contiguous one in one instant, the ratio of space and time will always be equal to one. Moreover, the speed of any continuous motion will always be the higher speed ever reachable by a mobile, because motion cannot take less time than one instant to cover one punctual space.

If a point is moved over three points, it will be moved continuously over this space in a time composed of three indivisibles. And it doesn’t mat-ter how much the motive power is increased, it cannot be moved faster, because continuous motion is the fastest.96

Crathorn doesn’t deny that there can be variations of speed, but they must be understood by discontinuity of motion. As an example, an indivisible can move over three punctual spaces through six instants. There are times of rest during the changing of places that sensitive cognition cannot grasp.97 Several examples are given, and Crathorn seems to be very proud of this thesis about speed when affi rming that

95 This sort of theory existed in the Arabic tradition. For a brief overview of these Arabic atomist theories, cf. Pines, Beiträge zur islamischen Atomenlehre; Wolfson, The philosophy of the Kalam; Baffi oni, Atomismo e Antiatomismo nel Pensiero Islamico; Jolivet, La théologie et les Arabes. On the extension of atoms in the Mutakallimun, see Dhanani, The physical theory of Kalam. On al-Ash’ari in particular, cf. Gimaret, La doctrine d’al-Ash’ari, I, 1 et III.

96 In Sent. q. 3, p. 256: “Si punctum moveatur super trium punctorum, movebitur continue super illud spatium in tempore composito ex tribus indivisibilibus. Et quan-tumcumque virtus motiva augeatur, non potest moveri velocius, quia motus continuus est velocissimus.”

97 Ibid. p. 256: “Si vero ponatur quod punctum motum moveatur per spatium trium punctorum discontinue, hoc potest contingere multipliciter: uno modo sic quod quiescat in quolibet puncto spatii per duo instantia, alio modo per tria, alio modo per quattuor et sic de aliis numeris, et talis motus potest sic velocitari. Unde si esset virtus motiva in tali gradu quod moveret punctum per spatium trium punctorum in sex instantibus, alia virtus quae esset dupla respectu primae, moveret id punctum per spatium trium punctorum in tribus instantibus . . . Sed apareat nobis quod multi motus sunt continui, cum tamen non sunt continui propter quietes interceptas, quas non percipimus.” Cf. also p. 258.


he has proved this point during the year of redaction of his Questions on the Sentences.98 Actually, we must limit Crathorn’s self-conceit about this paternity. Indeed, it might be noticed that this theory of speed is not Crathorn’s privilege and that he could have been aware of some version of it from elsewhere.

Very early in the Middle Ages, the Arabic theologians of the Kalam developed several different atomist theories. Some of them were really similar to Crathorn’s own view with regard to the importance attached to place, position and extension in the defi nition of atoms.99 More strik-ing is the fact that some of them had the very same idea about the speed of a continuous motion: variations of speed have to be explained by times of rest in motion.100 Of course, this kind of analysis of speed directly fl ows from the atomist view of time and space, whoever devel-ops it. But could Crathorn have known such an Arabic source? It is diffi cult to decide, but we know that this theory was available in the Latin West quite early thanks to the Latin translation of Maimonides’s Guide of the perplexed, in which he criticized in detail the Mutakallimun’s analysis of speed.101

In the Dux perplexorum, Maimonides summed up the atomist theory of the Mutakallimun in twelve points, among which the existence of simple

98 In Sent. q. 16, p. 456: “Ex dictis patet evidenter una conclusio, quam probavi isto anno, scilicet quod omnis motus vere continuus est velocissimus et quod implicat contradictionem unum motum vere continuum esse velociorem alio motu vere con-tinuo.” Crathorn probably refers to a university lecture different from his Questions on the Sentences or to some still undiscovered text.

99 See footnote 95 for bibliographical references.100 Cf. Ibrahim & Sagadeyef, Classical Islamic Philosophy, pp. 88–94. D. Gimaret gives a

detailed description of this theory (op. cit., pp. 114–115): “De ce que le parcours d’une distance est compris comme le franchissement obligé de toute la série de ‘lieux’ consti-tuant cette distance, tous ‘lieux’ d’égale dimension (il s’agit, en fait, d’atomes); de ce qu’à cette série uniforme de ‘lieux’ correspond une série pareillement uniforme d’instants d’égale durée (atomes de temps), chaque ‘franchissement d’un lieu’ correspond à un nouvel instant; de tout cela il résulte que tous les mouvements sont quantitativement égaux. Il n’y a pas de mouvement plus lent ou plus rapide qu’un autre, tout mouvement est déplacement d’un point-atome de l’espace au point-atome contigu, et cela dans le temps d’un instant-atome . . . Si l’un est plus lent que l’autre (c’est-à-dire parcourt dans le même temps une distance moindre), c’est que ses mouvements sont entrecoupés d’arrêts imperceptibles à l’œil, pendant que l’autre mobile continue d’avancer.”

101 On the reception of Maimonides in the Latin West, Cf. Kluxen, “Maimonides and Latin Scholasticism” and idem “Maïmonide et l’orientation de ses lecteurs latins”. For Aquinas’s reading of this Latin version, see Anawati, “Saint Thomas d’Aquin et les penseurs arabes: les loquentes in lege Maurorum et leur philosophie naturelle”. On the Latin version of the text itself, see Freudenthal, “Pour le dossier de la traduction latine médiévale du Guide des égarés.”


substances, of void, of instants of time, of qualifi ed atoms. Atoms are thus considered as small parts of the thing.102 Contrary to Crathorn, they seem to have held that atoms are unextended, but the conclusion concerning the variations of speed are all the same.103

Crathorn never quotes Maimonides, but we know that he was exten-sively used by other English scholars since Adam Marsh and Thomas of York.104 Interesting enough is the fact that we also fi nd the same explanation in Nicholas of Autrecourt whose dependence on Mai-monides has already been suggested by Andrew Pyle and Christophe Grellard105 and also in John Wyclif ’s Logica.106 In any case, Crathorn was not infl uenced by Autrecourt nor by Wyclif, since it is not possible historically speaking. Although we cannot decide if Crathorn’s has been infl uenced in some way by this Arabic theory, notice should be taken, however, of the existence of a matrix of arguments which cannot be reduced to the reappraisal of Aristotle’s critiques, nor to the simple reconstruction of Democritus through Aristotle. John E. Murdoch states that this position is shared by almost all the indivisibilists,107 but in 1330 Autrecourt and Crathorn were probably the fi rst ones to put it so clearly. Moreover, this kind of theory requires the mereotopological structure of the continuum previously studied, which wasn’t theorized by all the indivisibilists.

102 Maimonides, Dux seu director neutrorum sive perplexorum [Paris, 1520], I, 73, f. 32v (collation with the Ms. Sorbonne 601 (S), f. 38ra): “Ratio igitur primum [the separate substances exist] est: dixerunt siquidem quod mundus universaliter, id est omne corpus quod est in eo, est compositum ex partibus valde parvis que non habent partes . . .”

103 Maimonides, Dux seu director neutrorum sive perplexorum [Paris, 1520], I, 73, f. 33r (collation with the Ms. Sorbonne 601 (S), f. 38rb): “Dixerunt quod motus est mutatio substantiae separatae de numero atomorum unius, scilicet singularis substantiae, ad aliam substantiam propinquam et ex hoc sequitur quod non est unus motus velocior alio. Et secundum hanc positionem dixerunt quod vides [S: in] duo mobilia pervenire ad duos terminos diversos in remotione in eodem tempore non est quia unus motus sit velocior alio, sed causa eius est quia mobile cuius motus dicitur tardior habet plures quietes in spacio suo quam illud quod dicitur velocius . . . Quod autem credis moveri motu continuo aliquid contingit ex errore sensuum et brevitate in apprehendendo . . .” Note that in the text the substantiae separatae correspond to simple substances, i.e. to atoms, and not to angels in this context.

104 See Kluxen, “Maimonides and Latin Scholasticism” and idem, “Maïmonide et l’orientation de ses lecteurs latins.”

105 Cf. Pyle, Atomism and its Critics, pp. 336–337 and pp. 687–688; Grellard, “Les présupposés méthodologiques de l’atomisme: la théorie du continu chez Nicolas Bonet et Nicolas d’Autrécourt.”

106 John E. Murdoch has once remarked on the similarity between the three authors. Cf. Murdoch, “Atomism and Motion in the Fourteenth Century,” p. 52.

107 Cf. Murdoch, “Atomism and Motion in the Fourteenth Century,” ibid.


Let us now direct our attention to another chapter of physics in Crathorn’s thought: rarefaction and condensation. Unfortunately, generation and corruption are never explicitly analysed, nor alteration, even though we can imagine what could have been such an analysis in an atomistic fashion. Nonetheless, one can fi nd detailed but surprising developments on condensation and rarefaction.

10. Condensation and Rarefaction: The Limits of Crathorn’s Mereotopology

As we have seen, the role attributed to the notion of place is far more important in Crathorn’s physics than in Odo’s, Bonet’s or Autrecourt’s atomistic philosophy. The true fi nite divisibility of a continuum, accord-ing to Crathorn, is the fi nite divisibility of space: there are indivisible places where atoms can be located. One of the conclusions that should be posited from what has been said is the coincidence of an indivisible place with an indivisible thing. One atom, one single place. This can be traced in the different steps of his argumentation, for it is presupposed in the analysis of contiguity, continuity, motion and speed. Consider-ing this mereotopological framework, there are two possible ways of explaining condensation and rarefaction: these natural phenomena can be reduced to some kinds of alteration or can be understood as a change in the arrangement of atoms that occurs thanks to vacuum (empty spaces newly fi lled out, or in the contrary, new empty spaces in a body that cause an increase in size). Of course, one should hold a combined theory with both elements.

The fi rst way has been followed by John Wyclif, a few years after Crathorn,108 on the same mereotopological basis. It is now little known that Wyclif subscribed to an atomist theory of matter, combined with an Aristotelian hylomorphic world view.109 Unfortunately, recent scholars didn’t examine closely his principles, which are basically the same as Crathorn’s. Let us compare Wyclif ’s strategy to Crathorn’s in order to evaluate both of their solutions.

The fi rst element to be noticed is that Wyclif treats the problem of the continuum in a logical treatise (the Logicae continuatio, tractatus tertius,

108 Around 1363. Cf. Thomson, The Latin Writings of John Wyclif, p. 6.109 Cf. E. Michael’s chapter on Wyclif in this volume. See also Kretzmann, “Continua,

Indivisibles and Change” and Pabst, Atomtheorien des lateinischen Mittelalters.


ch. 9), when he turned to the study of local propositions, as ‘Socrates runs where Plato runs”. A few lines later, he abandoned this proposi-tional analysis for a long digression (132 p.) on the locus of the world and the situs of its atoms. According to Wyclif, there are punctual places110 as well as instants of time, and therefore atoms of matter and atoms of time. As for Crathorn, locus is nothing but the thing located,111 and thus quantity is only determined by its place. As a corollary to this, it is suggested that points occupy punctual places and can be located this way. But Wyclif ’s theory of situs is a bit more precise than Crathorn’s, for he defi nes the place of an atom within a defi nite landmark formed by the centre of the world and its poles.112 Wyclif ’s atoms resemble Crathorn’s indivisibles because they occupy a single place in the world but also because they are considered as qualifi ed entities.113 In his Logicae continuatio, Wyclif even imagines, against a sceptical theologian, that no one would refuse the possibility for God to create a punctual substance in each punctual space; therefore, the existence of punctual substances is possible.114 Moreover, these punctual substances exist in a fi nite number in the universe.115

This description of atoms seems very similar to Crathorn’s, as well as for its physical implications: contiguity and continuity can be defi ned in terms of contiguity of places116 and a continuous motion

110 Wyclif, Logicae continuatio [Dziewicki], p. 2: “Et sic, quamvis species situs punctualis sit principium integrandi omnem situm divisibilem, tamquam minimum metrum illius generis, tamen totalis situs mundi est nobis mensura cognoscendi alios sitos particu-lares . . .”

111 Ibid. p. 3: “. . . omnis situs est aliquid situari.”112 Ibid. p. 4: “Ideo, sicut in natura omne motum vel mobile innititur alicui fi xo,

sic non est possibile nos locum cognoscere, nisi in comparatione ad aliquod fi xum. Sicut ergo mundus ad eius motum situalem presupponit polos et centrum quieta, sic presupponit ad eius situacionem eadem . . .”

113 Actually, as Emily Michael shows in her chapter in this volume, there are dif-ferent ontological levels of atoms: unqualifi ed and unextended indivisibles, elemental atoms and minima naturalia.

114 Logicae continuatio [Dziewicki], p. 33: “Similiter, ut credo, nullus theologus negaret quin Deus de potentia absoluta potest facere substantiam punctualem . . .; et tunc patet quod punctualitas vel punctus que est substancia huiusmodi esse punctualis, est actus positivus [in] illa substancia . . . Punctus ergo potest esse. Nec dubium quin situs essent correspondenter iuxtapositi, cum situs sit subiectum situari.”

115 Ibid. p. 36: “Unde impossibile est quod aliquis numerus substanciarum vel punc-torum vel aliud preter deum sit simpliciter infi nitum.”

116 Ibid. pp. 30–31: “Similiter de immediacione ubicacionum vel situum indivisibilium, patet quod est dare tales immediatas. Nam est dare duo puncta immediata, ut patet de corporibus tangentibus, sic ubicaciones vel situaciones eorum sunt immediate. Et cum illi situs manent expunctantes alia puncta, patet quod quandocumque alter eorum erit


has the higher speed ever reached by a mobile.117 There is at least one important difference between Crathorn and Wyclif, because the latter seems to deny the existence of vacuum. Moreover, Wyclif is not clear about the quantity or extension of atoms, saying sometimes that they are unextended and sometimes that they occupy a place and are real parts of things118. What about condensation and rarefaction now?

Wyclif ’s answer is astonishing. The world is totally full of matter:

It is to be noticed that the world is composed of certain atoms, and can neither be increased nor diminished, nor locally moved in a straight line, nor be fi gured in some other way (. . .).119

This point requires some explanations. There are three premises to this conclusion:

1) there is no vacuum2) there is a fi nite number of atoms and places in the world3) two things (even indivisibles) cannot be in the same place at the same

time (i.e. two atoms cannot exist in the same punctual place)120

The fi rst conclusion to be drawn is that all possible punctual places are fi lled by atoms.121 The second is that a body cannot be increased or diminished in size without losing or gaining parts.122 The quantity of

occupatus aliquo punctuali intra corpus, reliquus erit occupatus punctuali sibi imme-diato. (. . .) Similiter de instantibus; videtur quod erunt immediata . . .”

117 Ibid. p. 97: “Ad illud dicitur quod impossibile est aliquod indivisibile velocius moveri localiter quam continue in quolibet instanti dati temporis describere situm suum punctualem.”

118 See Michael’s chapter for the texts.119 Ibid. p. 1: “. . . notandum mundum componi ex certis athomis, et nec posse maiorari

nec minorari nec moveri recte localiter vel aliter fi gurari . . .”120 Ibid. p. 42: “ad tertium dicitur quod impossibile est multa puncta vel substancias

punctuales esse simul in eodem situ indivisibili.”121 Before being informed by substantial forms and arranged by aggregation, the

prime matter is composed of indivisibles which fi ll all possible places. Cf. Logicae continua-tio [Dziewicki], p. 119: “. . . ymaginandum est igitur unam essenciam corpoream, in principio productam, esse ex indivisibilibus composita, et occupare omnem locum pos-sibilem, nec esse secundum eius partem aliquam corruptibilem, nisi forte per divisionem vel separacionem unius partis a reliqua.” On the notion of prime matter in Wyclif, cf. Kaluza, “La notion de matière et son évolution dans la doctrine wyclifi enne.”

122 Ibid. p. 69: “Ex istis colligitur quod nullum corpus potest esse maius aut minus quam prefuit, nisi propter adquisicionem aut deperdicionem materie, quamvis putatur quidlibet rarefactum esse maius quam prefuit, ignorando situs quod perdit intrinse-cus, sicut et ignoratur commutacio situum extrinsecorum pro intrinsecis in partibus condensati.”


matter always remains the same. Indivisible parts of the whole world can change from one punctual place to another by an extrinsic force, but without occupying an empty space. A sponge, when wrung, has less air than before, but air is not vacuum according to Wyclif.123

Even if he rejected assumption 1), Crathorn should have contended that the quantity of matter is constant in the world for he at least admits 2), that the number of atoms is fi nite and that to one punctual space must correspond one indivisible. He thus could have explained condensation and rarefaction by the absence of interparticulate spaces in an absolutely dense body and, on the contrary, the presence of some empty punctual spaces in a rarefi ed body (the second way previously described).124 He also could have thought, as Wyclif did, that conden-sation and rarefaction are kinds of alteration. But Crathorn did not agree either with Wyclif ’s solution or with the second one just posited, as Nicholas of Autrecourt did for example. He attempted to overturn these argumentations by denying 3) and claiming—with some appar-ent self-contradiction—that several points can be in the same punctual place.

In this view, a rarefi ed body will have a few parts existing in one and the same place; on the contrary, a dense body will have a lot of parts existing in the same place.125 In this case, to what extent can a body be diminished or increased? Crathorn’s answer is quite radical:

A point can be rarefi ed, because there is nothing more in the notion of a point than to be something located and situated, having no parts outside of parts, which is compatible with having parts existing in parts and being composed out of several things really and essentially distinct, though these are not distinct according to place and position. Thus: everything that has several parts, indistinct according to position and place, can be rarefi ed; therefore it is not incompatible for a point to have several parts indistinct according to place and position; therefore it is not incompatible for a point to be rarefi ed . . . A point can be rarefi ed into a body, for it is posited that a point has a hundred parts indistinct according to position, in such a way that it is composed of a hundred parts: such a point can

123 Cf. p. 63.124 Nicholas of Autrecourt held such a theory. Cf. Christophe Grellard’s chapter

in this volume.125 In Sent. q. 14, p. 421: “Corpus esse rarum non est aliud quam illud corpus non

habere partes aliquas loco et situ indistinctas vel habere paucas partes situaliter vel localiter indistinctas . . . Unum vero corpus esse densum non est aliud quam illud corpus habere multas partes localiter indistinctas vel in eodem loco et situ.”


be rarefi ed into a body, because with rarefaction, the parts of the point can be distinguished according to their position . . .126

This also works the other way around for condensation. If parts are distinguished according to their position in a body, after condensation they will not be further differentiated by their position.127

This new element is threatening for the coherence of Crathorn’s view, because all his arguments rest on the correspondence between an indivisible and a punctual place. As we have seen, the position of an indivisible in space implies the notion of a single place (locus punc-tualis), but place also determines the quantity of the located thing; and then points or indivisibles must have in some way a certain quantity. Crathorn presupposed this view of indivisibles throughout his analysis of the part/whole relation, contiguity, continuity, motion and speed. Moreover, his fi nitist view of the number of indivisibles seems to imply, as in Wyclif, the principle of correspondence between punctual places and indivisibles. Indeed, from this consideration about rarefaction and condensation, the fi nite number of atoms seems to be postulated rather than demonstrated by Crathorn, for nothing seems to prevent infi ni-ties of indivisibles if more than one indivisible can occupy a punctual place and rarefaction could be in principle infi nite.128 So, according to this view, the world could be densifi ed into one point or rarefi ed into an infi nite space, which is contradictory with what we have posited before.

Although Crathorn endeavoured to restrict his analysis to fi nite cases, always giving examples with a fi nite number of indivisibles,129 his fi nit-

126 Ibid. p. 423: “Punctum potest rarefi eri, quia non est plus de ratione puncti, nisi quod sit aliquid situatum vel locatum non habens partem extra partem, cum quo stat quod habeat partem inexistentem parti et quod sit quid compositum ex pluribus realiter et essentialiter distinctis, licet omnia ista sint loco et situ indistinctas. Tunc sic: omne id quod habet plures partes situaliter vel localiter indistinctas potest rarefi eri; sed non repugnat puncto habere plures partes situ et loco indistinctas; igitur non repugnat puncto rarefi eri . . . Punctum potest rarefi eri in corpus, qui ponatur quod punctum habet centum partes situ indistinctas ita quod componatur ex centum partibus; tale punctum potest rarefi eri in corpus, quia partes talis puncti possunt per rarefactionem situ distingui . . .”

127 Ibid. p. 424: “. . . corpus potest condensari in punctum. Si enim punctum posset rarefi eri in corpus, eadem ratione idem corpus condensari potest in punctum, etc.”

128 It is a commonly held argument in the Middle Ages against indivisibilism. See, for example, its use in the dispute between John Buridan and Michel of Montecalerio (cf. Celeyrette’s chapter in this volume).

129 In Sent. q. 14, p. 424: “Vigesima prima conclusio est quod punctum rarefi t naturaliter per actionem agentis naturalis, et hoc patet sic: sicut alias probavi, corpus


ist physics becomes impossible to prove. What is more, indivisibles are no longer quantifi ed, if they can be a hundred in the same indivisible place, and they also cannot be parts, because P1 and P2, which are the basic claims of Crathorn’s indivisibilism, require that indivisibles be considered as parts of the continuum, and thus as real and actual things occupying a punctual place. Wyclif saw the diffi culty and strongly affi rmed the principle of equivalence between the number of punctual places and atoms.130 So, we might say that he is incoherent or that he cannot escape from the mathematical and infi nitist view of indivisibles considered as mere unextended mathematical points. The second alter-native wouldn’t fi t well with many of his developments. Therefore, it seems more likely that he is a little incoherent.

On the other hand, we may assume that the main reason for this departure from the basic mereotopological principles is motivated by theological reasons. Indeed, we must keep in mind that Crathorn not only treats the issue in q. 3, but also in the more theological context of other sorts of indivisibles, as God, Angels and souls (q. 15 for example). The questions of God’s ubiquity and Angel’s motion were the most frequent pretext in the Middle Ages when dealing with indivisibles and their location.131 In this respect, Crathorn was forced to affi rm that the

componitur ex superfi ciebus, superfi ciebus ex lineis, linea ex punctis, et ita omne corpus fi nitum componitur ex punctis fi nitis. Si igitur a multitudo maxima et fi nita omnium punctorum unius corporis situ distinctorum et totalium et suppono quod illud corpus rarefi at, quousque secundum omnem dimensionem habeat quantitatem dimensivam, quae sit centupla respectu quantitatis dimensivae primo habitae, et suppono quod omnes partes illius corporis ante rarefactionem sint uniformiter et aequaliter densae et quod uniformiter et aequaliter rarefi ant. Istis suppositis sequitur quod facta rarefactione multitudo punctorum ipsius corporis rarefacti accipiendo puncta modo praedicto, scilicet quorum quodlibet situ distinguitur ab alio et quorum nullum est pars alterius puncti, multitudo talium punctorum est centupla respectu multitudinis punctorum consimiliter acceptorum ipsius ante rarefactionem. Igitur multa puncta, quae facta rarefactione sunt localiter distincta, fuerunt ante rarefactionem indistincta localiter et tamen per suppositum quaelibet pars vocando partem totam rem occupantem partem spatii ante rarefactionem sit aequaliter rarefacta.”

130 Wyclif, Logicae continuatio [Dziewicki], pp. 42–43: “. . . impossibile est multa puncta vel substancias punctuales esse simul in eodem situ indivisibili . . . Unde argumenta hominum volencium destruere quotlibet talia puncta in eodem situ indivisibili petunt pro fundamento quod non sit possibilis composicio continui ex non quantis.”

131 The continuum is frequently treated in the second book of the Sentences when dealing with angels. John Duns Scotus, for example, took this pretext in his different versions of his commentary on the second book of the Sentences (e.g. Ordinatio [Balic e.a.], d. 2, q. 5, pp. 296: “Utrum angelus possit moveri de loco ad alium locum motu continuo”). All the same, in his Reportatio super Sententias [Etzkorn e.a.], II, d. 2, q. 3, Walter Chatton asks “Utrum motus componatur ex indivisibilibus” and introduces


soul can be everywhere in the body, though it is indivisible. Same here for God and Angels, who are purely indivisible but can be in different places, either extended or punctual. Finally, it is plausible that Crathorn tried to fi nd a general defi nition of condensation and rarefaction that would suit all sorts of indivisibles, material and spiritual. Incidentally, he did not seem to notice the incoherence of his views on density and rarity with the basic principles of his atomism.

11. Conclusion. A Theory of MINIMA NATURALIA?

In most of the medieval debates on the nature of the continuum and the existence of indivisibles, one has to choose between a geometrical view of indivisibles—which does not require such a detour through mereotopology—and a more physical atomism, which requires at some point to consider indivisibles as sharing some physical properties (mini-mal quantity, qualities or nature, position in space, etc.). If one chooses the second option of the alternative, as Crathorn did with many others in the fourteenth century, it cannot be asserted at the same time that several indivisibles can occupy the same punctual place and that indi-visibles are in fi ne defi ned by the place they occupy as parts of something. Beyond the simple topological intuitions of Crathorn’s atomism is the thesis that indivisibles are actual parts of the continuum. The idea of minimal actual parts was already in Aristotle when he accepted the existence of minima naturalia. Indeed, he accepted both claims, that a continuum is infi nitely divisible in potentia, but fi nitely divisible in actu. The result of an actual division of a continuum into its parts would be a minimum of fl esh or bones in the case of human beings. If we continue the division, parts won’t exist any more: they won’t have any nature, nor quantity, nor virtue or action, etc. But isn’t this idea what Crathorn claims about indivisibles? They have a nature, a quantity and

his development as follows: “Et quia non potest sciri de motu angeli utrum sit con-tinuus vel discretis in motu nisi sciatur utrum motus et alia continua componantur ex indivisibilibus, ideo quaero propter motum angeli utrum quantum componatur ex indivisibilibus sive permanens sive succesivum.” Gerard of Odo’s discussion about the continuum occurs in the context of God’s and angels’ simplicity. Cf Super primum Sententiarum, dist. 37, Mss Naples, Bib. Naz. VII. B.25, ff. 234v–244v; Valencia, Cated. 139, ff. 120v–125v: Ad quorum evidentiam querenda sunt quatuor . . . Tertium utrum motus angeli habeat partem aliquam simpliciter primam. See also Gregory of Rimini, Lectura super secundum Sententiarum [Trapp], d. 2, q. 2, pp. 277–339: “Utrum angelus sit in loco indivisibili aut divisibili.”


they can move and act, as actual parts of the continuum. Moreover, they can exist separately as Aristotle’s minima naturalia. Of course, the last considerations about condensation and rarefaction tend to make the description of indivisibles quite confused. Finally, are there two distinct theories in Crathorn? A theory of atoms conceived as kinds of minima naturalia on the one hand, taking up some of Aristotle’s intuitions, and a theory of extensionless points on the other hand.

John Wyclif, for example, who held a very similar theory of indi-visibles, clearly distinguishes between the two levels. According to him, there are unqualifi ed and unextended indivisibles and, at a higher level, they are minima naturalia, which are composed of such primitive indivisibles, already joined in elemental atoms.132 The problem is that Crathorn makes no distinction between unextended indivisibles (as points) and qualitative atoms (kinds of minima naturalia). Moreover, he always uses a geometrical vocabulary, as the majority of his contem-porary fellows. For example, he sometimes refers to “points of gold” that occupy a punctual place. Consequently, it becomes impossible to distinguish between the level of points and the level of minima naturalia, because both seem to have a nature and a kind of extension.133 Crathorn would probably have to admit the Wyclifi an theory. In any case, except for theological contexts, Crathorn’s atoms are a sort of mix between Aristotle’s minima naturalia and Democritus’s atoms.

In this paper, we tried to show how systematic is the use of mereol-ogy—relations of parts and wholes—and topology—notions of place and position—in Crathorn’s atomism. In fact, his whole atomistic conception of the world rests on these notions. We then compared him with some of his contemporaries in order to demonstrate his originality, although his point of departure was probably Harclay’s and Chatton’s positions. Indeed, Crathorn was probably the fi rst real atomist in Oxford who considered indivisibles as real and actual enti-ties. As we have already mentioned, the origin of his atomism could be

132 See Michael’s chapter in this volume.133 When Crathorn distinguishes between mathematical and corporeal fi gures, he

seems to assert that they share the same properties. Sent. I, q. 3, p. 239: “Dicetur forte quod lineae istae, quas nos facimus et videmus, sunt corpora; et ideo ex coniuctione et distantia talium linearum, quae veraciter non sunt lineae, non potest argui coniunctio vel distantia verarum linearum. Contra: istud non est bene dictum, quia si istud esset verum, numquam geometria fuisset inventa nec unquam posset doceri; non enim potest doceri sine talibus protractionibus.” We thus may infer that points and corporeal indivisibles have the same properties.


traced back to his interpretation of some of Aristotle’s physical works, and notably of the De generatione et corruptione and the De caelo, and also to his readings of other ancient sources as the Liber sex principiorum, the Dux perplexorum of Maimonides or the De elementis of Isaac Israeli. A thorough examination of his sources would shed light upon some other points in the theory, for Crathorn knew a lot of authors to whom he does not always refer.134

The general conclusion that can be drawn from this reading of Crathorn’s Questions on the Sentences is that there are no hints of math-ematical concern in such a physical theory. Of course, Crathorn endeavoured to respond to the rationes mathematicae that he knew from John Duns Scotus, but only as a kind of formal obligation. Crathorn’s goal was fi rstly to fi nd some physical principles to describe natural sub-stances and in the process to fi nd an alternative ontology to Aristotle’s metaphysics. This attitude becomes evident if one turns to his critique of Aristotle’s categories, where it is affi rmed that the distinction of the ten categories does not make sense. Thus, it may be helpful to read Crathorn’s texts on indivisibles in the light of his metaphysical claims, in order to understand what he took and what he rejected from Aristotle. In any case, Crathorn’s is a good example of an atomist theory which is not merely geometrical, as John Murdoch used to say about all the indivisibilist positions of the fourteenth century. On the contrary, he attests the existence of different atomist traditions and of the use of different kinds of ancient sources, not only aristotelian ones.

134 We may assume that Isaac Israeli’s De elementis, which has been translated into Latin by Gerard of Cremona at the end of the twelfth century, is one possible source for this kind of mixed theory that has been described in the previous paragraph. Crathorn refers to this text in another context (In Sent. q. 4, p. 280: “Quid autem sit risus, dicit Isaac libro suo De elementis.”). In the second part this book, Isaac Israeli presents is position against the atomists. He asks whether Galen’s assumption that there must be minimal parts in bodies, Aristotle’s assertion that there must be minima naturalia, Democritus and the Mutakallimun atomist theories, could fi t together in a unifi ed conception. Cf. Isaac Israeli, De elementis. In Isaac Israeli opera omnia (Lyon, 1515), f. 7D sqq. This point would deserve another paper.

AN INDIVISIBILIST ARGUMENTATION AT PARIS AROUND 1335: MICHEL OF MONTECALERIO’S

QUESTION ON POINT AND THE CONTROVERSY WITH JOHN BURIDAN

Jean Celeyrette

The debate concerning the nature of point and the divisibility of the continuum between John Buridan and a master named M. of Monteca-lerio1 has been known for a long time.2 It is described as such in one of the tables of the ms. BNF Lat. 16621 (f. 195r)3 in which both masters’ texts appear. This scientifi c manuscript comes from Etienne Gaudet’s collection.4 It is the only surviving witness of Montecalerio’s question and it is particularly diffi cult to transcribe. This is why, despite several attempts, the text still remains unpublished, even though Buridan’s text, which appears in another manuscript, has been published by Zubov.5 Although Bernd Michael considered any number of possiblities concerning the identity of Buridan’s oppponent, until recently he was

1 In the manuscript, the master is called Montescalerio. Following W.J. Courtenay I will designate him as Montecalerio (see footnote 5).

2 K. Michalski thought that Buridan’s text was directed against Walter Burley. Cf. Michalski, La physique nouvelle et les différents courants philosophiques, p. 120. Indeed, we will se below that Montecalerio’s position is close to Burley’s.

3 “Deinde sequitur determinatio de puncto, videlicet utrum sit alica res addita linee, Magistri Johannis Bridani contra magistrum de Montescalerio que durat circuiter per 7 folia [. . .] deinde determinatio magistri M. de Montescalerio de puncto, videlicet an per divisionem continui corrumpatur aliqua res, contra magistrum Johannem Bridan per XI folia usque in fi ne sisternum.” (BNF Lat. 16621, f. 195r).

4 On Etienne Gaudet’s collection, see Kaluza, Thomas de Cracovie, Contribution à l’histoire du Collège de Sorbonne.

5 Zubov, “Jean Buridan et les concepts du point au XIVe siècle.” The edition is based on the manuscripts Paris, BNF Lat.16621, ff. 196r–202v and 203v, and Paris, BNF Lat. 2831, ff. 123r–129v. Zubov’s edition is reliable even if a comparison with Montecalerio’s text enabled me to make some corrections. The quotations hereafter will be either drawn from Zubov’s edition, noted QP, or from the manuscript Paris, BNF Lat. 16621 noted EG. In this last case, for reasons of clarity, I indicate by (M.M.) or (Bur.) the author of the text which I quote. I will use Montecalerio’s presentation of Buridan’s arguements and the references will be thus with EG; I will also indicate the corresponding passages in QP.

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very much a mystery.6 In two recent papers, William Courtenay shed a new light on Michel of Montecalerio, master of arts of the French Nation around 1342 and still active in 1346.7 Most notably, we learn that he was of outstanding importance, an element that could explain why Buridan devoted such attention to his writings.

A comparative reading of both texts reveals that the controversy, which seems to be quite severe by the tone of their respective retorts,8 should be situated in the context of the Parisian discussions about Ockhamist physics. We know that Buridan does not share Ockham’s view about the nature of point in the two known versions of his commentary on the Physics.9 Nevertheless, he accepts it in the text we are commenting upon in this chapter. Indeed, like Ockham, Buridan maintains that “point” is a privative name and that a point, as every privation, is nothing real: as an example, a terminative point ( punctum terminans) corresponds to the fact that a line goes until there and not beyond. But this means nothing else than the word “point” is the signifi -cate of the proposition “a line goes until this limit and not beyond”.10 It seems that Buridan shares this view in this context, even if he does not in others.

On several occasions Montecalerio refers to this Buridanian position to reject it but his detailed refutations are lost: the end of his question which was certainly devoted to this refutation has disappeared. How-ever, it should be noted that what remains of Montecalerio’s text fi lls 9 folios while 11 folios are announced in the table at the f. 195r. So the lost refutation was at most two folios long. As Buridan’s exposition

6 Michael, Johannes Buridan: Studien zu seinem Leben, seinen Werken und zur Rezeption seiner Theorien im Europa des späten Mittelalters, pp. 451–452.

7 Courtenay, “The University of Paris at the Time of Jean Buridan and Nicole Oresme” (especially pp. 8–10) and idem, “Michael de Montecalerio: Buridan’s opponent in his quaestio de puncto.” The text of Montecalerio, that of Buridan’s question mentioned in footnote 12, and other unedited texts from the ms Paris, BNF Lat.1 6621 are in the course of publication by Zénon Kaluza and myself.

8 See, for example, Montecalerio’s reply: “Et miror de isto doctore sic experto quod istius <quod> exposuit fuit immemor.” EG (M.M.) ff. 222v–223r.

9 Cf. Celeyrette, “La problématique du point chez Jean Buridan”. We fi nd a clear statement of Ockham’s position in his Quaestiones in libros Physicorum, q. 63 and in his Tractatus de quantitate. The detailed quotations and references appear in the above-men-tioned paper, from note 27 to 30, pp. 93–94.

10 About the notion of complexe signifi cabilia, see Gabriel Nuchelmans, Theories of the Propositions. Ancient and Medieval Conceptions of the Bearers of Truth and Falsity. On Buridan’s refutation of the Ockhamist position in his Questions on the Metaphysics and his Sophismata, see pp. 243–250. See also Biard, “Les controverses sur l’objet du savoir et les complexe signifi cabilia à Paris au XIVe siècle.”

an indivisibilist argumentation at paris around 1335 165

of his own opinion takes only 1 folio out of 7, that is to say 5 pages out of 32 in Zubov’s edition, we can be sure that we still have access to most of the debate.

1. The Controversy

Before going through the analysis of Montecalerio’s texts, we must make clear who is responding to whom. At fi rst sight, it seems that each responds to the other; the arguments of a venerabilis doctor are mentioned or quoted in Buridan’s text and said to be contra me, all of which are also found in Montecalerio’s question and conversely.

However, a more careful reading allows us to settle this point. In Montecalerio’s question all of Buridan’s arguments—or nearly all of them—are cited, repeated, disproved, and usually in the same order that they appear in Buridan’s text, so that we can reasonably think that Montecalerio’s text is responding to Buridan’s. As to Buridan’s quota-tions of Montecalerio, it can be noted that they are less close to the text than in the latter’s quotations of Buridan. Then to what do they cor-respond? Montecalerio probably supplies the answer, for he refers—at least six times—to an alia determinatio (or alia quaestio) that he claims to have written in reply to Buridan’s dicta, the existence of which is con-fi rmed by the incipit of Buridan’s question: Montecalerio is mentioned in this text as a doctor who was opposed to some Buridanian teachings on the nature of point.11 Buridan was not the only master aimed at, for concerning his opponent Buridan talks about “those against whom he is disputing.”12 We can thus think that in the text we have in hand, Montecalerio quotes—sometimes textually—Buridan’s refutations of the arguments taken from the “other determination,” that he gives his contra-objections and confi rms his preceding arguments.

Thus, the controversy may be reconstructed in this way: fi rst, teach-ings by Buridan, and very likely others, defending positions close to Ockham’s; then, Montecalerio’s determination against those teachings,13 not written or lost; then, Buridan’s response as copied by Etienne Gaudet

11 “Doctor unus venerabilis obviavit quibusdam dictis meis de puncto multum subtiliter.” QP p. 63, l. 1–2.

12 “illi contra quos ipse disputat” QP p. 73, l. 25.13 Montecalerio indicates that he is also the author of a treatise De motu which is

lost.

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which would be the text edited by Zubov; fi nally, Montecalerio’s text, the one we are discussing here, also copied by Etienne Gaudet.

Nothing here allows us to maintain that this controversy corre-sponds to a publicy head disputatio.14 Moreover, the absence of ritual polite formulas and the lively tone assumed by both men makes such a hypothesis improbable.

The different steps of the controversy cannot be precisely dated. I hold the weak hypothesis that both texts are contemporaries of another question disputed by Buridan and copied by Etienne Gaudet in the same manuscript, and to which they can be compared. It is entitled “On the possibility that one and the same thing exist and do not exist at the same time” (De possibilitate existendi eandem rem et non existendi simul in eodem instanti ) and is dated from 1335.15 William Courtenay, in the above-mentioned paper, shows that Montecalerio was master of arts around 1340; therefore, it is reasonable to think that the controversy took place between 1335 and 1345.

2. Montecalerio’s Text

Montecalerio’s text comes as a quaestio the plan of which is the following:

Question: is a thing corrupted by the division of the continuum(Utrum per divisionem continui corrumpatur aliqua res: EG ff. 214r–223v)

First Article

First Part of the First Article ( ff. 214r–214v)Buridan’s arguments corresponding to QP pp. 63–73.

– A continuing point ( punctum continuans) would be neither in potency nor in act. (f. 214r)

– The introduction of terminating points ( punctum terminans) is in vain. (f. 214r)

– A point would be neither a substance nor an accident. (f. 214v)

14 Cf. Weijers, La ‘disputatio’ dans les Facultés des arts au moyen âge, pp. 46–47.15 EG (Bur. De possibilitate existendi . . .) ff. 233v–237r. The question ends as follows:

“Explicit questio de possibilitate existendi rem eandem et non existendi simul in eodem instanti determinata per magistrum Bridam anno domini millesimo CCC°XXX°V°. Deo Gratias.”


– The introduction of points distinct from bodies is useless, since all appearances can be saved without them. (f. 214v)

Second Part of the First Article ( ff. 214v–217v)Answer to the four preceding arguments and confi rmation of the arguments from the alia determinatio with a series of propositions and corollaries.

– Reply to the fi rst argument (ff. 214v–216v). I study this answer below.

– Reply to the second argument (f. 216v): points are the cause of the line.

– Third reason (ff. 216v–217v): the existence of points is not naturally impossible.

– Fourth reason (f. 217v): points do exist and this is supported by the classical example of the sphere tangent to the plane.

Second Article

In this section Montecalerio’s three reasons in support of his thesis—i.e. that a point is a real accident—are presented. These are the three rea-sons that Buridan refutes.16 It follows that they have been put forward in the alia determinatio.

First Part of the Second Article ( ff. 217v–220v)Defence of the fi rst reason: if a point didn’t exist, it would follow that the division of a continuum would amount to a local motion of this continuum (the parting of its sections), a position that has been refuted in the alia determinatio.17

– Refutation of a Buridanian thesis maintaining that the division of a continuum is an intermediary indivisible change between rest and local motion formed by the parting of sections (ff. 217v–218v). I present this argumentation below.

– Division consists in a change.

16 “Tunc volo defendere rationes meas quibus probavi punctum esse accidens quas iste doctor multum reprobat et non veraciter impugnat.” EG (M.M.) f. 217v.

17 “Arguebam enim primo sic quod sequeretur quod divisio seu discontinuatio esset motus localis huius continui, et falsitatem consequentis probavi diffuse in alia determi-natio.” EG (M.M.) f. 217v.

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– Division is neither an alteration nor an increase in size, nor a decrease in size, nor a generation (f. 218v).

– Division is not a local motion. Montecalerio then concludes that division is a corruption, for it cannot be another type of change (argument of suffi cient division) (ff. 218v–220r).

– Refutation of Buridan’s objections against this position (f. 220v).

Second Part of the Second Article ( ff. 220v–222v)Defence of the second reason: if a point didn’t exist it would follow that a non-continuous thing would be continuous.18

– Presentation of Buridan’s arguments: the cause of a body’s division into parts is not the surface but the nature of the body (f. 220v).

– Refutation of these arguments and contra objections (ff. 220v–222v)

I give a general survey of this argumentation below.

Third Part of the Second Article ( ff. 222v–223v)Defence of the third reason: the surface of a body exists because it is what prevents us from seeing through it.

Last Part of the Second Article ( f. 223v)Discussion of Buridan’s position according to which “point” is a privative name. This discussion is interrupted immediately after the presentation of Buridan’s position.

It is obviously impossible to give an exhaustive presentation of such a lengthy and complexly stuctured discussion. We will here restrict ourselves to select some of the arguments indicated above. As Montecalerio’s text is the last in the series, the examples of arguments presented here will always be preceded by those from Buridan’s to which Montecalerio is replying and which, for that matter, he expounds very thoroughly. Of course, it is impossible to know precisely what appeared in the two preceding steps.

18 “Secunda ratio mea talis est: sequeretur quod non continuum esset continuum (ms: contiguum) etc.” EG f ° 220v. The correction from «contiguum» to «continuum» is justifi ed by Buridan’s text: “non continuum esset continuum” QP p. 82, l. 1–2. For Buridan, this reason is one of the main arguments against his adversary.


Reply to the First of Buridan’s Arguments, that “a Continuing Point can neither be in Potency not in Act” (Second Part of the First Article)

As every potency can be actualized, the argument would amount to denying the possibility for a continuing point to be actualized. This is a classical topos of the anti-indivisibilist argumentation and its discussion is one of the longest in Montecalerio’s text (ff. 214v–216v).

Presentation of Buridan’s Argumentation ( f. 214r)

Montecalerio’s presentation of the Buridanian argumentation—espe-cially of the objections that he has made against some arguments of the alia determinatio—is particularly detailed. Maybe this is due to the fact that the initial reasoning, the one that appeared in the preceding step, that is to say in Buridan’s dicta, was marred by rough error and that Buridan maintains the same erroneous reasoning in QP.

Buridan formulates four suppositions assumed, he says, by his adver-saries:

– If a continuing point is in act, it is the case for all the others.– The continuing point really exists in act.– Points are ordered in a continuum in such a way that none of them

is equally distant from the terminating point.– The terminating point can be separated by imagination, or even in

reality, from the rest of the continuum.

This being stated Buridan reasons as follows. Let us separate this fi rst point, which is possible by the fourth supposition. Either there exists a fi rst point or not.

– If it is the case, this new fi rst point is in act according to the second supposition and there is no intermediate point between the fi rst and the second point; and from the fi rst supposition every other point of the continuum is in act. Applying the same reasoning to this new point, it appears that there is another point equally in act immediately after it.

– If it is not the case, one should admit that in a fi nite space there are several things existing in act and ordered by their position without one being before the other. Now, he briefl y says, that there is an infi nity of things does not prevent that there should be a beginning

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in existing things, ordered in relation one to the other, and, moreover, it would be contradictory that there be none.19

The same reasoning is taken up again in various classical forms, in particular that of the point of contact of a sphere rolling along a line on a plan; it is shown that after the fi rst point of contact there is a second, etc.

The adversary is then led to admit that immediately after each point there is another, which is denied in Physics VI20 and refutes the initial assumptions.

The second part of Buridan’s reasoning (“si non etc.”)21 constitutes one of the many alternatives of an argument whose paternity is usu-ally allotted to Harclay, that is to say to an indivisibilist. It is worth noting that in this context this argument is used to refute the existence of indivisible points. With regard to the form of the argument, the existence of a fi rst point in the multitude of points in the continuum from which the terminating point has been removed appears in Harclay and his successors or critics as the result of the fact that God, who sees the totality of the points of the multitude, necessarily sees the fi rst of them.22 In Buridan’s argument, no allusion is made to the divine absolute power. This is why the parallel is perhaps more relevant with another indivisibilist text: Gerard of Odo’s Quaestio de continuo.23 Gerard thinks in the same way as Buridan, but he considers the multitude of

19 “His suppositis probatur consequentia quia: vel est aliquid ipsorum ante omnia, aut nullum. Si aliquid, tunc inter ipsum et primum punctum nullum est medium punc-tum et sic erunt proxima. Si nullum est ante omnia alia, hoc est inymaginabile quod cum in alico spacio fi nito sut plura ordinata secundum situm et actum ibi existentia et tamen nullum sit ante alia. Non apparet ratio, ymo contradictionem implicat, quia infi nitas in mediis non tollit primitatem in ordinatis extra se invicem existentibus ergo etc.” EG (M.M.) f. 214r; cf. QP p. 64, l. 15–27.

20 Aristotle, Physics VI, 231a21–231b18.21 The argument runs as follows: if not, one would have to accept that there are

many things in act and ordered by their respective positions in an infi nite space without one being before the others.

22 “Certum est quod Deus modo intuetur omne punctum quod possit signari in continuo. Accipio igitur primum punctum in linea inchoativum linee; Deus videt illum punctum et quodlibet aliud punctum ab isto in hac linea. <Tunc a primo puncto> usque ad illum punctum immediatiorem quem Deus videt intercipit<ur> alia linea aut non. Si non, Deus videt hunc punctum esse alteri immediatum. Si sic, igitur cum in linea possint signari puncta, illa puncta media non erant visa a Deo.” Henry of Harclay, “utrum mundus poterit durare in eternum a parte post,” quoted in Murdoch, “Henry of Harclay and the Infi nite,” p. 228, note 24.

23 On this text, see Sander W. de Boer’s contribution in this volume.


the parts of the continuum, not the multitude of points. He then shows the existence of “fi rst” parts, thus indivisible, of which the continuum is made up. However, it is necessary to introduce some shade into what precedes because it is not completely obvious that Gerard considers that his reasoning must be applied to magnitudes as magnitudes and not to natural things.24

Buridan, on the other hand, says it clearly. Moreover, considering that the points are infi nite in number, he affi rms very clearly that this infi nity does not prevent the existence of a fi rst point: the formula he uses, reported twice by Montecalerio, is infi nitas non tollit primitatem. The error is all the more remarkable because if I have reconstructed the polemic correctly, it appeared in his fi rst teaching and his adversary had then replied to it: QP refers to the objections of Montecalerio.25

3. The Polemic about Buridan’s Reasoning

It does not appear that Montecalerio appreciated the full implications of his critique of Buridan and there is no evidence that he posed theo-retically the question of the existence of an infi nite multitude and that of a possible fi rst element. His principal objection is built on a parallel with parts: if one considers the proportional parts of a continuum, starting from the fi rst point and having them ordered according to the time a mobile would take to go through them, Buridan’s inference

24 Gerard of Odo, Quaestio de continuo, Ms. Madrid, Bibl. Nac. 4229, f. 180rb.: “Tertia ratio est haec: in omni ordine essentiali est aliquid simpliciter primum respectu cuius omnia dicunt posteriora, quia alias non esset ordo. Sed totius ad suas partes est ordo essentialis, quia essentialis dependentia et in via constitutionis et in via resolutionis. Ergo in illo ordine est aliquid simpliciter primum, puta aliqua pars vel partes, quae sic est pars quod non est totum. Continuum autem est quoddam totum consitutum ex partibus suis. Ergo in ipso est aliqua pars quae sic est pars quod non est totum. Et sic ipsa erit indivisibilis. Quare continuum componetur ex primis partibus indivisibilibus, quod est propositum.”

25 “Contra istam rationem obicit reverendus doctor dicendo quod per consimilem rationem probabit de partibus continue proportionalibus . . .” QP p. 65, l. 16–17. It is known that this mode of reasoning, in any case in the form which Harclay gives to it, is refuted by Wodeham. One of Wodeham’s refutation consists in saying that when the fi rst point of a continuous line is removed, it remains a line without a fi rst point: “Illa ratio est bona ad improbandum puncta, sicut tangetur in questione proxima. Tamen, sustinendo puncta indivisibila, dici haberet quod destructo solo puncto terminante lineam, linea manet interminata, nata tamen terminari, et ideo <linea> privaretur termino, id est puncto. Nec tamen propter hoc esset infi nita positive, sed solum privative, id est carens fi ne nato haberi.” Adam de Wodeham, Tractatus de indivisibilibus, [Wood], p. 108, l. 5–10.

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seems to apply; and if this inference were true, one should admit the existence of a fi rst proportional part, which is obviously false. The argument, presented as a resumption of the alia determinatio, appears in various forms. It seems indisputable and Buridan’s response becomes even more surprising.

Buridan essentially challenges the analogy between points and pro-portional parts. He affi rms that this parallel doesn’t work, because the fi rst point is reached in fi rst position after the terminus, whereas the fi rst of the proportional parts, if it existed, would be reached in last posi-tion by successive divisions starting from the whole.26 The only analogy which he regards as valid is the one between points and parts in which a mobile would move successively, and in the same order, starting from the terminus, exactly as it would do for the points; then, he says, it is true to affi rm that one part is closer to the terminus because if the continuum is divided into two halves, one will be closer than another; into four quarters, one quarter will be closer than the others, etc.27

In a lengthy response to these arguments, Montecalerio distinguishes between two types of parts: those in which a part is always the part of another, and those in which no part is part of another, in other words, consecutive parts.28 His refutation of Buridan’s replies and the confi rma-tion of his former argumentation are based on this distinction.

Moreover, Montecalerio uses his adversary’s argument in a polemical way to establish the existence of points; in fact, he uses approximately the same reasoning as did Gerard of Odo. It is enough, Montecalerio says, to consider the sum of the consecutive parts: there is an infi nite number of parts in relation to a terminus. Buridan’s inference, Monte-calerio contends, demonstrates that given this infi nity of parts, there would be one part closer to the terminus. This part cannot be divisible. If it were, then one of its halves would be closer than the other. It must,

26 “Et addit iste doctor quod non similiter arguitur hic (pour les parties proportion-nelles) et de punctis quia in istis propter ordinem concluditur primum ex ea parte qua proceditur in infi nitum, sed in punctis concluditur primum ex ea parte qua incipitur.” EG (M.M.) f. 214r; cf QP p. 67, l. 30–34.

27 “Sed inter quartas est una prior omnibus aliis quartis, et inter octavas est una prior omnibus aliis quia prima, ergo sic debet esse in punctis quia primo est aliquid propinquius omnibus aliis.” EG (M.M.) f ° 214r; cf QP p. 65, l. 21–66, l. 3.

28 The length of the development on the two families of parts is a priori surpris-ing. But, apparently, at this time in the Parisian milieu these considerations were not regarded as trivial. It is notably the case in Burley.


therefore, be indivisible, i.e. a point. Montecalerio then turns Buridan’s argument against its author.

4. Montecalerio’s Position

Montecalerio specifi es that the existence of points of this type, i.e. indivisible parts, does not correspond to his position. For him, the point is not a part, because points and parts have different natures. After a point there is always a part which is a line, and not another point, just as immediately after a line there is a point which is its terminus. Of course there is no point closer to the terminus, for this would imply that the continuum is composed of indivisibles, a position he does not admit. On the other hand, he does admit that a whole is different from its parts, the word “parts” meaning for him divisible parts.29

In his conclusion to this fi rst article, Montecalerio states his position without really developing it, a position that will be demonstrated more precisely in the following article: for him, to be divided or undivided is a property that belongs only to quantity. Therefore it cannot be said that the subject is divided by itself, it is divided through quantity. To explain this Montecalerio uses the example of extension (this parallel appeared in the other determination): if a subject is extended, it is through quan-tity; it is not extended by itself, for its extension comes from a process of rarefaction, i.e. by an alteration; it is therefore extendable. Finally, the point is an accident in the category of quantity, it is indivisible, it exists subjectively in a substance that is undivided, unextended in act but naturally divisible.30 Montecalerio is particularly clear about this. There is no punctual substance, no indivisible atom, and the fact of being punctual and without extension is only accidental. Of course, Buridan retorts that it supposes the possibility of an infi nite rarefac-tion, a point which is conceded by Montecalerio. Let us notice that

29 It is interesting to make a link between this aspect of Montecalerio’s position and some conclusions to the fi rst questions in the book VI of Burley’s In Physicam Aristotelis expositio et quaestiones.

30 “Et hoc voluit Aristoteles primo Physicorum quod divisio et indivisio primo con-veniunt quantitati et aliis non nisi postea, et quia punctum est de genere quantitatis et est simpliciter indivisum et inextensum, ideo non habent a subjecto, sicut nec quantitas, divisionem, sed ipsum dat subjecto suo indivisionem et inextensionem actualem, sicut quantitas divisionem. Per hoc potest dici ad formam rationis, sicut prius, quod punc-tum est accidens indivisibile existens subjective in substantia actu indivisa et inextensa, divisibili tamen naturaliter.” EG (M.M.) f. 217r.

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the objection traditionally opposed to the composition of a continuum by indivisibles, according to which a point added to a point does not make something larger, does not impact this position.

5. The Argument of the Infinite Multitude

Let us go back to the arguments concerning infi nity. One can only be struck by their awkwardness. Neither of the two adversaries seems to consider that there should be the slightest diffi culty in speaking of an infi nite totality. There is no trace of a logic of the infi nite: expressions as “categorematic infi nite” and “syncategorematic infi nite” are missing. More surprising is Buridan’s stubbornness in the maintening a bad argu-ment despite of the acceptable objections of his adversary, even if it is also used by other Masters, like Gerard of Odo, to obtain a indivisibilist conclusion immediately. A way of saving Buridan’s argument would be to imagine that Buridan wanted to turn against his adversaries an argument which he knew to be false. But nothing in his presentation nor in Montecalerio’s reply supports this charitable interpretation. However, one can certainly take into account the fact that developing refutations or conter-objections at this point in his question, Buridan deviates from his own problematics. This could explain the lightness of his answers before an adversary that he sometimes—and more likely for polemical purposes—pretends not to take seriously.

Nevertheless it is indisputable that in this text Buridan does not at all seem in control of the concept of infi nite: he does not express any hesitation when speaking about an infi nite multitude and he regards it as a totality having the same properties as a fi nite totality. We are very far from the treatise on the infi nite in the last version of his Physics.31

6. The Demonstration of the Existence of Points (First Part of the Second Article)

This demonstration apparently fi gured in the alia determinatio, so that there too Montecalerio’s argumentation is mingled with answers to Buridan’s objections to the former series of arguments.

31 As an example, question 17: “utrum omni numero est numerus maior supposito semper quod nullum continuum est compositum ex indivisibilibus.” ed. J.M.M.H. Thijssen, Johannes Buridanus over het oneindige, pp. 55–69.


His reasoning is as follows. On the basis of the fact that division, into two for example, is a change in the subject, he establishes successively that this change is neither an increase, nor a decrease, nor an altera-tion, nor a generation, nor a local motion. From this, he deduces that it can only be a question of corruption which is instantaneous. This enables him to affi rm that there is a thing which is corrupted, a thing which is the point of continuity.32

This demonstration is preceded by the refutation of the Buridanian position on the division of a continuum. I will outline this refutation, because several of the arguments given in response to Buridan are taken up later to establish that division is not a local motion, which is in fact the node of his reasoning.33

7. Buridan’s Argumentation, its Refutation by Montecalerio (ff. 217v–218v) and the QUAESTIO DE POSSIBILITATE EXISTENDI . . .

Buridan’s position about the division of a continuum is the following: division is indeed a change but it is nothing more than a relative local motion of parts, one of which remains motionless while the other one is moving, or both of which are moving in contrary directions. This change, like any local motion, is not instantaneous but temporal. The diffi culty of this position is that since the body was continuous before its separation, no distinction is made between continuous parts and con-tiguous parts, and thus between rupture of continuity (discontinuatio) and rupture of adjacency (discontiguatio). In order to overcome this diffi culty Buridan considers that between the rest of an undivided continuum and the local motion of separation of parts, or more generally between a rest and a local motion, both temporal, there is an indivisible change which makes the parts which were continuous become contiguous parts. But Buridan specifi es at once that when one speaks about this indivis-ible change one is only saying that at fi rst the parts were not divided and that afterwards they are divided.34

32 A similar reasoning can be found in Burley, notably in his Tractatus de formis [Down Scott], pp. 70–71. For the similitude between Montecalerio and Burley about the nature of point, see Alice Lamy, Substance et quantité à la fi n du XIII e siècle et au début du XIV e siècle. L’exemple de Gautier Burley, p. 270.

33 Cf. QP pp. 74–78.34 “Ad formam igitur rationis dicit quod mutatio divisionis est indivisibilis quia iste

terminus «divisio» signifi cat «partes prius immediate fuisse non divisas et immediate

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Montecalerio, who considers that division is not a local motion and must refute Buridan’s position, shows that between rest and motion there is no indivisible change. He does this with three proposals. This part of the controversy, like the preceding one, is paradoxical. What exactly is the indivisible change introduced by Buridan? Either it does not have any reality and there is no longer difference between contigu-ous and continuous anymore, or it is real and then it seems that one is led to admit the existence of instants. Now it is Montecalerio, the indivisibilist, who refutes a position seeming to imply the existence of indivisible instants, whereas this position is supported by Buridan, for whom points and instants do not exist.

The need for the introduction of an indivisible change between two incompossible temporal things is the subject of Buridan’s question de possibilate existendi . . . to which I have already referred,35 and which can be regarded as a treatise on the fi rst and the last instant of motion. Let us clarify. Imagine two incompossible things: a black body that is whitening and the same body that is simply maintaining its whiteness, i.e., that is at rest. The intermediate instant will be characterized by the fact that the body will be at the same time white and not white, therefore two contradictories will be true at the same time. However it will be in the sense given above, i.e., immediately before the body was black, and immediately after it will be white. At the end of the question Buridan gives an interpretation of this simultaneity in terms of measures with an example in which he supposes that a non-white at rest is transformed into white and then remains white. It is necessary then, he says, to suppose that there exist fi ve measurements, namely three moments and two instants: the three moments measure the rest of the white and of the non-white and the motion of whitening, the two instants measure the intermediate instants.36 It is worth noting

post istas esse divisas et separatas—dans QP: discontinuas-. Et idem est quod signifi -cat «unam partem non moveri sine altera immediate prius, et immediate post unam moveri sine alia vel ambas moveri motibus contrarie» [. . .] Secundum solvit quod secundum veritatem mutatio divisionis non est indivisibilis, scilicet generatio vel cor-ruptio indivisibilis, sed est motus localis divisibilis quia una pars totius removetur ab altera secundum profunditatem et hoc non potest esse in instanti.” EG (M.M.) f. 218r; cf QP p. 77, l. 12–20 et p. 78, l. 10–11.

35 Cf. footnote 15.36 “Et illorum oportet esse quinque mensuras, scilicet tria tempora et duo instantia:

primum tempus mensurat non esse albi in sua quiete et secundum tempus in suo fl uxu ad esse tempus mensurans generationem albi, tertium tempus mensurat esse albi in sua quiete; instans quod est medium inter tempus generationis et tempus quietis precedentis


that Buridan gives here a certain reality to the instant of intermediate change, since he regards it as a measurement, that of the instant during which the two incompossibles are simultaneous.

This last question of Buridan being dated, as we said, from 1335, it can perhaps give us an indication for dating this controversy. In his presentation, Buridan indicates that he argued elsewhere that two incompossibles are possible at the same instant and that many people had been surprised by this and that it was for this reason that he wrote his question.37 It is tempting to connect it with Montecalerio’s observa-tion made in his text:

And my third reason, given in the other question [the alia determinatio], proves this point, namely that it is only because of a succession of time that a proposition can be true now and its contradictory be true afterwards, and this doctor [Buridan] does not formally reply to that.38

The succession of times being opposed to the succession times/instants, the drafting of the question de possibilitate existendi . . . could be, inter alia, Buridan’s response to Montecalerio, which could give a terminus ante quem to our controversy.

Even if one does not accept this conjectural reconstruction, it is worth noting that the problematics, the references and a number of arguments that appear in the question de possibilitate existendi . . ., are coherent with what is found in the polemical text of Buridan. That all these questions were composed at roughly the same time is very probable.

8. The Common Surface of Parts as a Formal Cause of their CONTINUATIO (Second Part of the Second Article)

The continuity of the parts of a line consists in having a common extremity, the continuity of the parts of a body in having a common surface. As Montecalerio considers that the extremity point and the

mensurat momentum a quo incipit motus vel generatio, et instans medium inter tempus generationis et quietem sequentem mensurat ultimum momentum generationis.” EG (Bur. qu. de possibilitate existendi . . .) f. 236v.

37 “Multi mirantur qualiter alias dixi quod possibile sit idem esse et non esse in eodem instanti, propter quod intendo dare medium per quod non solum ostendetur hoc esse possibile sed necessarium.” EG (Bur. qu. de possibilitate existendi . . .) f. 233v.

38 “Et hoc probat ratio mea tertia quam feci in alia questione, scilicet unum contra-dictorium non potest nunc esse verum et post aliud nisi ratione successionis temporum, ad quam iste doctor in forma non respondet.” EG (M.M.) f. 222r.

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common surface are accidents in the category of quantity, he allots logically to this point or this surface the status of cause, formal but non material, of the continuity of the corresponding parts.39 Buridan’s objection is, therefore, that if the cause of the continuity of the parts is an accident in the category of quantity, one cannot explain why it is easier to divide a piece of cheese than a stone of the same quantity, i.e. of the same volume, or why it is easier to put in continuity two parts of water than two stones of same size, etc. And then, Buridan affi rms that the cause of setting two things in continuity (continuatio) is not in the category of the common quantitative accident which would be a common surface, it can be only qualitative and/or substantial. Many examples are given: that fl uid bodies are easily put in continu-ity, especially when their moisture is not viscous, whereas it is not so easy for others, which are not fl uids. Buridan explains this by the fact that the parts of the fl uid body interpenetrate easily whereas those of the other objects do not completely touch.40 Buridan’s argumentation takes a very physical turn, with the evocation of experiments which are not only thought experiments, but experiments that he describes as essential for philosophical knowledge. He even invokes the activity of the alchemist.41

To this formidable objection Montecalerio answers with a succession of four propositions, and reaffi rms that the causes of the continuity of a body are indeed its common parts and their surfaces and not some agreement (convenientia) between substantial or qualitative natures.42

His arguments do not take the same quasi experimental form; they are repeated in various forms, and remain very general:

1) In the Eucharist the separated quantity is continuous, and yet there is no substantial or qualitative nature present.

39 There too a comparison with Burley is relevant.40 “Et ideo causa continuationis <non> est nisi substantialis vel qualitativa vel

u traque: quia corpora bene fl uxibilia faciliter continuantur et specialiter si non sit humiditas viscosa, sed alia non fl uxibilia non continuantur, et causa est quia prima bene subintrant se invicem, secunda nullo modo.” EG (M.M.) f. 220v; cf QP p. 81, l. 1–15.

41 QP p. 81, particularly l. 19–34.42 “Quarta propositio est quod causa suffi ciens continuationis <1°> non est conve-

nientia in natura substantiali; <2°> quod nec convenientia in qualitatte vel similitudo; 3° quod nec ambo sunt causa suffi ciens continuationis.” EG (M.M.) f. 221r.


2) If a quality causes a continuity, it must be humidity, because certain humid bodies, continuous, heated, are transformed into ashes, which are discontinuous. However, humidity is not the cause of continuity since fi re is continuous.

3) The sky is continuous and yet it does not have a corruptible quality.4) Contrary to what Buridan says, when parts of water are brought

closer they are not immediately in continuity, but are only contiguous. They interpenetrate at once and because of this interpenetration, surfaces are corrupted, and a new common surface is generated which ensures continuity.

5) A piece of dead wood can be in continuity with a piece of alive wood although these two pieces of wood have different substantial natures.

In conclusion, to confi rm the position he had expounded in the other determination, he adds somehow maliciously that when Buridan con-tends that no power could divide continuum without a local motion or without an alteration, he contradicts the divine absolute power.43

Taking into account, however, the arguments of his adversary, Mon-tecalerio introduces the distinction between immediate formal cause of the continuation or division of parts of a continuum, and causes that make the continuation or division easier. He maintains that the surface (or the line or the point) common to the parts is the formal cause of continuation, and that the two parts ready to be put in continuity are its material cause. On the other hand, what makes continuation or division more or less easy varies according to the nature of the bodies we consider. Adapting certain arguments of Buridan he admits that this cause is sometimes qualitative: in the case of liquids, a nonviscous humidity can make easier the continuation or division of the parts; in

43 Buridan: “Ad illud autem quod adiungitur de potentia divina forte diceretur quod sine motu locali aut aliqua facta mutatione humidi continuantis Deus non posset discontinuare partes lapidis. De quo tamen nihil asserere intendo.” QP p. 82, l. 32–p. 83, l. 2. Montecalerio: “Sed respondendo ad alias, specialiter ad secundam, dicit quod per nullam potentiam poterat fi eri de continuo non continuum nisi facto motu locali vel alteratione, et quia salva eius reverentia hoc est contra potentiam divinam et forte contra naturam alico modo, ut probavi superius in defensione prime rationis, ideo ratio manet concludens necessario quod punctum est, et per consequens omnes rationes mee—celles de l’autre détermination-manent in suo vigore.” EG (M.M.) f. 222r.

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some cases, humidity can only make easier the division, and in other cases, it cannot cause either of them, etc.44

9. Conclusion

One will certainly feel frustrated after having read this text—or rather these texts. It teaches us Montecalerio’s position, but in an incidental way, and we do not really have an exposition in due form: only the answers to Buridan’s criticisms remain. The alia determinatio was supposed to give such a presentation, and perhaps it would have allowed us to know the nature and the extent of Montecalerio’s debt with regard to Burley. Given the present state of the texts, such a debt is only probable because of our very rudimentary knowledge of Montecalerio on the one hand, but also because of the disparate character and the lack of coherence of Burley’s argumentation on the other hand.

In addition, as it was pointed out earlier, the criticism of the Buridanian position, a position very close to Ockham’s, is missing in Montecalerio’s text. Such a criticism would have perhaps provided expla-nations regarding the point of view adopted at that time by Buridan on the status of quantity. It is clear that he departs from Ockham on this topic in the two known versions of his physics, but more clearly in the ultima lectura than in the tertia lectura.45 However this question becomes central in our debate on the nature of points since one of

44 “Patet quod alia est causa continuationis et divisionis prima et immediata et alia est causa facilitatis vel diffi cultatis continuationis et divisionis. Quia continuationis causa formalis et immediata est superfi cies vel linea vel punctus communis, sed causa materialis immediata sit quanta vel quantitates nate habere idem ultimum. [. . .] In aliquibus est humiditas fl uxibili, hoc est non coagulata bene nec bene permixta sicco, ut patet in liquidis, et hoc est causa facilitatis divisionis quia talis humiditas et licet possit tenere partes unitas in eodem ultimo, tamen faciliter cedit dividenti [. . .] In aliquibus autem talis humiditas non est causa continuationis nec satis discontinuationis et divisionis sicut in igne.” EG (M.M.) f. 222r.

45 In the tertia lectura which is generally dated from the beginning of the years 1350 Buridan seems to reply in a conservatory way in accordance with the via antiqua (quantity is an accident distinct from the substance) but without engaging clearly with the question utrum omnis res extensa est magnitudo (qu.I-7). On the other hand in the corresponding question of the ultima lectura (qu.I-8), after giving arguments favorable to the ockhamist position (the quantity is not really distinct from the substance), and refuting them, he declares that these refutations are insuffi cient and gives new ones which are presented as personal and which enable him to determine against Ockham that nulla substantia est magnitudo. On the two versions of Buridan’s physics, see my paper cited in footnote 9.


the protagonists considers that the point is an accident in the category of the quantity. Alice Lamy highlighted this point concerning Burley’s polemic against Ockham.46

Still, it remains that we have here, even incompletely reported, the example of a Parisian academic debate on Ockhamist physics, a debate which shouldn’t be neglected, all the less so since Buridan, a Master whose importance cannot be disputed, here defends the Ockhamist point of view. Moreover, it must be stressed that no other text on the continuum and no other text of Buridan’s than those I mention here appear in Etienne Gaudet’s collection.

What are the formal characteristics of this polemic? First of all, a great promptness of tone. An example has been given at the begin-ning of this article,47 but there are many other instances of this sort: Buridan does not hesitate to write that his adversary answers very weakly (valde debiliter); on several occasions he ironically admires the arguments of his adversary, while the other retorts that Buridan does not have to admire them, but that quite to the contrary it is he who admires—ironically—what Buridan claims to demonstrate, etc. The apparently respectful tone of the incipit is completely out of phase with the general tone of the two texts.

One is also struck by the lack of theoretical reference to the infi nite, which was pointed out earlier, and by the remarkably restricted place occupied by the logical considerations in Montecalerio, but also in Buridan who is known for having written logical treaties in the years 1320. I will not try to explain this fact, but will simply notice that when studying the divisibility of continuum, putting aside the initial argu-ment, the two protagonists call upon considerations on the rarefaction of the air or fi re, the transparency of the sheets of gold, but make no reference to the refi nements of the supposition theory, of the composite and divided sense, etc.

Last but not least, mathematics are mostly absent. They are men-tioned only once, when Buridan, indeed in a rather clumsy way, asserts that mathematics treat natural things from the point of view of the quantity and that if points—quantitative accidents of natural things—existed, there should exist mathematical points. The argument

46 Alice Lamy, Substance et quantité à la fi n du XIII e siècle et au début du XIV e siècle, l’exemple de Gautier Burley, pp. 228–277.

47 See footnote 8.

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is weak since points indeed exist in mathematics. Actually Montecalerio does not take the trouble to answer it. One fi nds no allusion, even to challenge their relevance, to the rationes mathematicae very fashionable among English scholars since Duns Scotus.

All these features of our controversy make it very different from English polemics like the one which opposed Chatton and Wodeham,48 but in the current state of our knowledge we cannot say up to which point it is characteristic of the Parisian debates of the years 1330.

48 See, for example, Murdoch & Synan, “Two Questions on the Continuum: Walter Chatton (?), O.F.M. and Adam Wodeham, O.F.M.”

JOHN WYCLIF’S ATOMISM1

Emily Michael

The fourteenth century evangelical doctor, John Wyclif (1320–1384),2 a prominent,3 if controversial, Oxford master, was an atomist. Like other scholastics of his time, Wyclif adopted a hylomorphic ontology, and he accepted the Aristotelian view that all corporeal things are composed of and reducible to four elements (earth, air, fi re, and water). Still, unlike his contemporaries, he also maintained that the body of the total universe, and each individual body within it, is composed of and reducible to a fi nite number of elemental atoms.4

John Wyclif is dubbed the evening star of scholasticism and the morning star of the Reformation by H.B. Workman, and by such contemporary commentators as Kenny and Spade.5 In another paper,6 I examined the question of whether he can with some justice also be dubbed the morning star of a reformation in science, for the fact is that his distinctive atomism as well as his mind-body dualism anticipates, in some respects, developments of early modern natural philosophy. The fi nal section of this chapter will provide a discussion of this distinctive

1 I gratefully acknowledge that research for this chapter was partially supported by a grant from the Research Foundation of the City University of New York. All translations are my own, unless otherwise noted.

2 The best biographical study of Wyclif ’s life is by Workman, John Wyclif, A Study of the English Medieval Church; Kenny, Wyclif, provides a clear and concise introduction to Wyclif ’s thought.

3 Conti, “Analogy and Formal Distinction: On the Logical Basis of Wyclif ’s Meta-physics,” p. 133, says that Wyclif “was one of the most important and authoritative thinkers of the late Middle Ages.” For an excellent bibliography, see: Thomson, The Latin Writings of John Wyclif: An Annotated Catalog.

4 See also, for discussion of Wyclif ’s atomism: Pabst, Atomtheorien des lateinischen Mittelalters; Zubov, “Walter Catton, Gérard d’Odon et Nicolas Bonet;” Kretzmann, “Continua, Indivisibles, and Change in Wyclif ’s Logic of Scripture.”

5 A point refl ected in Lewis Sergeant’s title John Wyclif, Last of the Schoolmen and First of the English Reformers. Workman, John Wyclif, A Study of the English Medieval Church i, 4: “The fi rst of the reformers was, in fact, the last of the schoolmen.” Kenny, Wyclif; John Wyclif, On Universals, [trans. Kenny], introduction by Paul Vincent Spade. See also Milton, Areopagitica [Bohn], ii, 91.

6 Michael, “John Wyclif on Body and Mind.”

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atomism. I will begin with an exploration of several other questions that Wyclif ’s atomism raises.

First, we might note that this distinctive and interesting atomism is now little known.7 What is more, though Wyclif was well known in his time, currently there is no evidence of discussion or infl uence of his atomism in the natural philosophy of his immediate successors. One might well ask why, though this fourteenth century atomistic view is clearly a striking development in natural philosophy, it received so little attention and had so little impact in its time and why this early atom-istic account, which is clearly a dramatic development in the history of thought, is now still so little known. That is:

Q1: Why the neglect of John Wyclif ’s atomism?

Second, one might reply to this query that the uniqueness and neglect of Wyclif ’s atomistic matter theory is easily understood in the light of the seeming lack of necessity for and perhaps inconsistency of an atomistic account in conjunction with a hylomorphic ontology. One might add that scholastic resistance to a Wyclifi an corpuscular matter theory was also supported by Aristotle’s strenuous rejection of atom-ism. This raises a second question, namely, what is the rationale for and the framework of an atomism that is coupled with a hylomorphic ontology. This, in fact, gives rise to two questions:

Q2: How is a corpuscular matter theory possible in conjunction with the Aristotelian view that prime matter and substantial form are the fundamental principles of corporeal things?

Q3: Why is a corpuscular matter theory needed in conjunction with the Aristotelian view that prime matter and substantial form are the fundamental principles of corporeal things?

In the following section, I will examine the fi rst two questions. In the subsequent section, in response to the fi nal question, I will consider, fi rst, the scholastic context of Wyclif ’s matter theory and, second, motivation for his commitment to an atomistic account. In the fi nal section, I will

7 See Spade, On Universals, introduction, p. 1: “John Wyclif ’s philosophical views are almost entirely unknown to modern scholars . . .” This is especially true of Wyclif ’s natural philosophy.

john wyclif’s atomism 185

further consider the rationale for and I will investigate the nature of John Wyclif ’s atomism.8

1. The Impact and Framework of Wyclif’s Corpuscular Matter Theory

1.1. Q1: Why the Neglect of John Wyclif ’s Atomism?

One might well ask, why, though Wyclif ’s fourteenth century atomistic view is clearly a striking development in natural philosophy, it seem-ingly generated so little interest among Renaissance philosophers and is now so little known. The fact is that though our Oxford master was an innovative thinker, a prolifi c writer, and extremely infl uential in his time, his philosophical views generally are little known even to his fi fteenth century successors. His biographer, Herbert Workman, tells us that, in Wyclif ’s lifetime, “his infl uence is beyond dispute. The source of this infl uence is clear. As a schoolman he was the acknowledged leader among his contemporaries at Oxford;”9 and Henry Knighton, though an implacable adversary, describes his foe as “the most eminent doctor of theology of his time, in philosophy, second to none, in the training of the schools without a rival.”10 Kenny correctly claims that Wyclif ’s philosophical views are as interesting and well-reasoned as those of Ockham or Scotus, and that his infl uence in his time was at least as extensive. This seems to argue that there was an interest in and infl u-ence of Wyclif ’s natural philosophy in his lifetime. What then is the reason for Wyclif ’s seemingly short shelf life and his current obscurity? One reason would seem to be the following.

Thirty-one years after Wyclif ’s death, the Council of Constance [1414–1418] declared him a heretic, and (after condemning theological and philosophical propositions attributed to him)11 ordered that his body

8 Works of Wyclif relevant to this study include: De Compositione Hominis [Beer], hereafter cited as DCH; De Materia et Forma [Dziewicki], hereafter cited as De M&F; Tractatus De Logica [Dziewicki] (Wyclif ’s matter theory is raised in his discussion of propositions of place to explain his distinctive notion of place, in vol. III, Logicae Continuatio, tract 3, ch.9, which will be hereafter cited as LC 3.9); De Ente [Dziewicki]; Trialogus [Lechler], also printed in 1525 and 1753, hereafter noted as Tr.

9 Workman, John Wyclif, A Study of the English Medieval Church., vol. i, p. 3.10 Ibid., vol. ii, p. 151.11 Conciliorum Oecumenicorum Decreta [Alberigo e.a.], p. 427. “Cum itaque in sacro

generali concilio nuper Romae celebrato decretum fuerit, doctrinam damnatae memo-riae Ioannis Wicleff damnandam esse, et libros eius huiusmodi doctrinam continentes

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be exhumed and that his corpse and all his works be burned.12 The Council dogmatically proclaimed that it forbids the reading, teaching, expounding and citing of the said books (they mention Dialogus, Trialogus, and the same author’s other books): “It forbids each and every Catholic henceforth, under pain of anathema, to preach, teach, expound, or hold the said articles or any one of them”13 and it “orders local ordinaries and inquisitors of heresy to be vigilant in carrying out these things . . .”.14 Jan Hus, the Evangelical Doctor’s infl uential Czech follower, though granted safe conduct to these proceedings by the Pope and the Emperor Sigismund, was condemned by the same Church council and burned at the stake.15 Nonetheless, Wyclif ’s works (copied, for example, by his Czech admirers and carried back to their homeland) survived, and one, his Trialogus, was printed in 1525.16 This provides some evidence

fore tamquam haereticos comburendos, et doctrinam ipsam damnatam et libros eius tamquam doctrinam insanam et pestiferam includentes, combustos fuisse cum effectu; et huiusmodi decretum huius sacri auctoritate concilii fuerit approbatum . . .”

12 For an excellent discussion of this condemnation and history of Wyclif ’s reputa-tion among Catholics, see Kenny, “The Accursed Memory: The Counter-Reformation Reputation of John Wyclif ”. See also Leff, “John Wyclif: The Path to Dissent,” and idem, Heresy in the Later Middle Ages. The focus of an offi cial Church Council on the condemnation of an individual and the establishment of this condemnation as Church dogma is unusual.

13 Conciliorum Oecumenicorum Decreta [alberigo e.a.] p. 422: “. . . inhibens omnibus ex singulis catholicis sub anathematis intermination, ne de cetero dictos articulos aut ipsorum aliquem audeant praedicare, dogmatizare, offerre, vel tenere.”

14 Ibid.: “Super quibus exsequendis et dibite conservandis mandat praedicta sancta synodus ordinariis locorum ac inquisitoribus haereticae pravitatis vigilanter intendere, prout ad quemlibet pertinet, secundum iura et coanonicas sanctiones.”

15 The Council declared that “John Wyclif, of accursed memory, by his deadly teaching, like a poisonous root, has brought forth many noxious sons, not in Christ Jesus through the gospel, as once the holy fathers brought forth faithful sons, but rather contrary to the saving faith of Christ, and he has left these sons as successors to his perverse teaching. This holy synod of Constance is compelled to act against these men as against spurious and illegitimate sons, and to cut away their errors from the Lord’s fi eld as if they were harmful briars, by means of vigilant care and the knife of eccle-siastical authority, lest they spread as a cancer to destroy others.” Ibid., pp. 426–427: “(Sententia contra Ioannem Huss) . . . vir damnatae memoriae Ioannes Wicleff sua mortifera doctrina non in Christo Iesu per evangelium, ut olim sancti patres fi deles fi lios genuere sed contra Christi salutarem fi dem, velut radix virulenta, plures genuit fi lios pestiferos, quos sui perversi dogmatis reliquit successores. Adversus quos haec sancta synodus Constantiensis tamquam contra spurios et illegitimos fi lios cogitur insurgere, et eorum errores ab agro dominico tamquam vepres nocivas, cura pervigili et cultro auctoritatis ecclesiasticae resecare, ne ut cancer serpant in perniciem aliorum.”

16 This work may have been known, for example, by J.C. Scaliger, who provides a rudimentary hylomorphic corpuscular matter theory in his Exotericae Exercitationes [Lyon, 1557], and, in turn, infl uenced such seventeenth century hylomorphic atomists as Daniel Sennert and Sennert’s student and advocate, Johannes Sperling.


of a role for Wyclif during the Renaissance. Nonetheless, the Council of Constance succeeded very well in suppressing Wyclif ’s views. All but one of Wyclif ’s massive array of philosophical works remained in manuscript form, unpublished until the late nineteenth century. Many remain unpublished to this day.

It might with merit be claimed that the evangelical doctor’s legacy was preserved, fi rst, through the martyrdom of Hus and his subsequent persistent and devoted following, and, second, through the propositions condemned. Some of these were clearly never asserted by Wyclif, for example, among the 45 recorded in session 8 of the Council, “God ought to obey the devil.” and, among the 58 recorded in the fi fteenth session, “Every being is everywhere, since every being is God.” Others distort features of his philosophic views. Of particular importance to the current study is the condemnation of such propositions as:

P1: “Any continuous mathematical line is composed of two, three or four contiguous points, or of only a simply fi nite number of points; and time is, was and will be composed of contiguous instants.”17

P2: “It must be imagined that one corporeal substance was formed at its beginning as composed of indivisibles, and that it occupies every possible place.”18

P3: “God cannot annihilate anything, nor increase or diminish the world.”19

This is a direct condemnation of foundations of Wylcif ’s atomism.20

Wyclif was declared a heresiarch by (as described by the Council, Session 15, 6 July 1415) “[t]his most holy general synod of Constance,

17 Conciliorum Oecumenicorum Decreta [Alberigo e.a.] p. 426 “51. Linea aliqua math-ematica continua componitur ex duobus, tribus, vel quatuor punctis immediatis, aut solum ex punctis simpliciter fi nitis; vel tempus est, fuit, vel erit compositum ex instan-tibus immediatis.”

18 Ibid.: “52. Imaginandum est, unam substantiam corpoream in principio suo ductam esse ex indivisibilibus compositam, et occupare omnem locum possibilem.”

19 Ibid.: “49. Deus nihil potest annihilare, nec mundum majorare vel minorare . . .”20 It might be conjectured that these propositions were condemned because of the

diffi culties atomism raises for an account of the Eucharist, in particular, in regard to the doctrine of transubstantiation. Of particular importance in Wyclif ’s downfall is the subject of fi ve condemned propositions, viz., his view of the sacrament of the Eucharist, which analysis was infl uenced by his atomistic account of the physical world. See, for example, Wyclif, De M&F (which is an early work), in the context of his discussion of substantial change, p. 189: “De conversione autem panis in corpus Christi, quam ecclesia vocat transubstanciacionem, est longus sermo, et mihi adhuc inscrutabilis.”

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representing the catholic church and legitimately assembled in the holy Spirit, for the elimination of the schisms, errors and heresies.”21 The lack of infl uence of his corpuscular matter theory can be explained by the general acceptance, among his contemporaries, of hylomorphism and of Aristotle’s arguments opposing atomism. But this does not explain the absence of all discussion of and seeming lack of knowl-edge about Wyclif ’s atomism. It seems likely that the repudiation and extensive condemnation of Wyclif ’s views, the prohibition “under pain of anathema” of the reading, teaching, expounding and citing of his works, and the order that his books, treatises, volumes and pamphlets be publicly burned had an impact on subsequent knowledge about his philosophical views and on his legacy generally.

1.2. The Ontological Structure of Material Substances

We turn now to the second question, that is, how can atomism be con-sistent with Aristotelian hylomorphism.22 Like other scholastics of his time, Wyclif accepted the view that prime matter and substantial form are the fundamental ontological principles of each material substance. What I wish to show is that Wyclif adopted a distinctive hylomorphic account, to be identifi ed in what follows as scholastic pluralism, and that scholastic pluralism is consistent with atomism.

Some commentators claim that Aristotelian hylomorphism is neces-sarily inconsistent with atomism. For example, Van Melsen says that, in Aristotle’s view, “If one admitted that some body was a compound, the logical conclusion had to be drawn that it possessed only one form.”23 But, in a hylomorphic atomism (like Wyclif ’s) “the original forms of the elements are said to remain [in the compound]. This impaired the unity of the compound in virtue of the rule: one substance means one form, and a plurality of forms means a plurality of substances.”24 Dijk-sterhuis, discussing Daniel Sennert’s hylomorphic atomism, similarly sees Aristotelian hylomorphism and atomism as inconsistent. He says that

21 Conciliorum Oecumenicorum Decreta [Alberigo e.a.], p. 421: “Sacrosancta Constantiensis synodus generalis, ecclesiam catholicam repraesentans, ad extirpationem schismatis errorumque et haeresium in Spiritu sancto legitime congregata, auditis diligenter et examinatis libris et opusculis damnatae memoriae Ioannis Wicleff per doctores et magistros studii generalis Oxioniensis.”

22 In what follows, hylomorphism will refer solely to Aristotelian hylomorphism. 23 Van Melsen, From Atomos to Atom: The History of the Concept of Atom, p. 126.24 Ibid., p. 125.


Sennert, “to reassure his philosophic conscience, introduces subordinate and higher forms, which had always been unanimously rejected by the scholastic philosophers.”25 What I wish to show is that these commenta-tors misrepresent the development of scholastic hylomorphism.

The fact is that from the inception of scholasticism in the thirteenth century until its demise in the late seventeenth century, Aristotelians were divided into two camps in their analyses of how prime matter and substantial forms compose material things. One hylomorphic approach, infl uenced by Thomas Aquinas and to be identifi ed as scholastic monism, is very familiar. It is this interpretation of form and matter that Van Melsen and Dijksterhuis identify as Aristotle’s own view. The other now little known approach, namely, scholastic pluralism,26 was associated in the thirteenth century with, for example, archbishops of Canter-bury, Robert Kilwardby and John Pecham (mentioned in this regard by Wyclif ),27 and, in the fourteenth century, was the interpretation of Aristotle’s view presented not only by Wyclif, but also by such prominent and infl uential fi gures as Scotus, Ockham, and John of Jandun.

Aquinas and his followers (i.e., scholastic monists) support the fol-lowing theses:

TT1: Prime matter is pure potentiality.TT2: Each substantial form is an absolute and immutable actuality,

determining a complete substance.TT3: An individual substance can have no more than one substantial

form, which inheres directly in prime matter.TT4: The one substantial form of a human being is a rational soul.

The scholastic pluralists maintain instead:

25 Dijksterhuis, The Mechanization of the World Picture, p. 283. Daniel Sennert was an infl uential seventeenth-century physician, who adopted atomism and a hylomorphic ontology.

26 For discussion of scholastic pluralism, see Zavalloni, Richard de Mediavilla et la controverse sur la pluralité des formes.

27 Wyclif, DCH, p. 74, cites Robert Kilwardby and John Pecham, respectively a Dominican and a Franciscan of the previous century, both of whom, as archbishops of Canterbury, issued condemnations in which the unicity of forms of Aquinas was seen as a heretical doctrine.

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TP1: Prime matter has an actuality of its own.TP2: Substantial forms are of two sorts, subordinate forms, which

determine a part of a substance or an incomplete substance, and supervening forms, which determine a total substance and place it in its species.

TP3: An individual substance can have a plurality of substantial forms.

TP4: An intellective soul is the ultimate supervening substantial form of a human being.

We fi nd here, in these respective theses, two very different interpreta-tions of Aristotle’s fundamental principles, prime matter and substantial form, and this, in turn, as might well be expected, had implications for a variety of other doctrines from embryology to immortality. So, for example, Aquinas and his followers maintain that each human being has a simple ontological structure. Each is wholly determined only by a rational soul that inheres in prime matter. Here prime matter is pure potentiality, incapable of existing apart from form, and since forms are the object of cognition, no idea of matter is possible, even for God. A substantial form gives a substance being, activates it, and makes it what it is. Socrates’s one substantial form is a rational soul, and his intellect is but one power of this soul, along with powers of nutrition, sensation, local motion, and appetite.

Scotus and Ockham support instead the view each human being has a pluralistic and hierarchical structure. From the viewpoint of both, Socrates is composed of prime matter, which has a positive reality of its own,28 and a plurality of kinds and grades of substantial forms.29 Both would agree that prime matter is the subject of the form of Socrates body, and that Socrates body, in turn, is the subject of a supervening substantial form, namely, Socrates human soul, which places him in his species and makes him what he is. But Scotus and Ockham each provide a different account of the corporeal and psychic substantial forms that determine a human being. In Ockham’s view, Socrates body

28 See, for example, John Duns Scotus on the reality of matter in Liber II Sententia-rum, dist. 12, q. 1, 2 [Vives], hereafter cited as Sent. See, also, Ockham’s discussion of matter, in William of Ockham, Summula Philosophiae Naturalis [Brown], OPh VI, pp. 181–195; and McCord Adams, William Ockham, pp. 680–690.

29 See, John Duns Scotus, Sent. [Balic e.a.], lib. IV, dist. 11, q. 3. See, also, Ockham, Quodlibeta Septem [Wey], 2nd Quodlibet, q. 11. See translation in Ockham, Quodlibetal Questions [trans. Freddoso & Kelly], vol. I, pp. 136–139.


(an incomplete substance) is determined by one subordinate substantial form. Scotus claims instead a plurality of corporeal substantial forms of the parts of Socrates’s body, e.g., of blood, bones, fl esh and the like. Further, each provides a different analysis of the relation between vegetative, sensitive, and intellective psychic principles. Scotus claims one soul in which these psychic principles are formally distinct. From Ockham’s viewpoint, Socrates must have two really distinct souls, an organic soul and mind. Jandun, taking yet another view, supports three really distinct souls, two inhering souls, vegetative and sensitive, and a mind, which, following Averroes’s monopsychism, is a separated substance that is one for all human beings. Scholastic monists adopted a single view, namely, that of Aquinas; scholastic pluralists developed a plurality of competing analyses of the structure of corporeal things.

Among Aquinas’s thirteenth-century contemporaries, the common view was the pluralistic one that at least some substances have more than one substantial form and that prime matter has a positive reality of some sort. Aquinas’s principle rationale for his opposition to plural-ism was a metaphysical claim that each substance must be one per se. Thomas argues: each entity is a single substance, but, since the actuality or nature of each substance is determined by its substantial form, if one subject has two substantial forms, it will have two actualities, and, therefore, it will be two substances.30 So, for example, Aquinas con-tends: if a man is a living thing by one form, the vegetative soul, and an animal by another form, the sensitive soul, and a man by another form, the intellective soul, it would follow that a man is not absolutely one.31 That is, a human being would, in fact, be three distinct things, a plant, for the vegetative soul is the plant soul, and a brute animal, as well as a human being. From this Thomistic viewpoint, distinct forms inform, i.e. determine, distinct substances. But a single substance must be one unifi ed entity; a substance cannot be composed of substances. Therefore, a single substance must have a single substantial form. Aquinas therefore claims:

TT5: The one substantial form of a substance is the principle that makes it one per se.

30 Aquinas, Summa Theologica, q. 76, art. 4.31 Aquinas, Summa Theologica, q. 76, art. 3.

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It was thereafter incumbent upon pluralists to respond to this meta-physical argument and demonstrate how a substance with a plurality of substantial forms is nonetheless one per se.

Aquinas’s rigorous and systematic monistic theory (TT1–TT5) was a controversial interpretation of Aristotle’s view,32 and there were numer-ous attacks against it. Perhaps the most important attack was William de la Mare’s 1279 Correctorium Fratris Thomae,33 a compilation of errors from the works of Aquinas in which William prominently includes TT1–TT5, as well as Aquinas’s view that the principle of individuation is signated matter.34 William ascribes Thomas’s complaint to an erroneous view of forms, in particular, to a denial of the relation of “grades of forms” within a single substance, and attributes to Aristotle the view that a substance in which there are various grades of forms is unifi ed by an ultimate form. William and subsequent pluralists maintained:

[TP5] The ultimate or supervening specifi c substantial form of a substance is the principle that makes it one per se.35

Further, the pluralists supported their claims by a vast arsenal of oppos-ing arguments, logical, metaphysical, physical, and theological.

The text of William de la Mare, a Franciscan, was adopted by the General Chapter of Franciscans in Strassburg on May 17, 1282, and was ordered, by the same Chapter, to be read along with the works of Aquinas as part of the text.36 This Franciscan endorsement of William de la Mare’s Correctorium further established, as common Franciscan

32 See Zavalloni (Richard de Mediavilla et la controverse sur la pluralité des formes, pp. 383–419) for a careful study of pre-Thomistic positions. Zavalloni provides convincing evidence that, prior to Aquinas, “la doctrine de la pluralité des formes . . . est univer-sellement acceptée en ce qui regarde le corps ou, du moins, le composé humain.” (p. 405)

33 This work appears in Correctorium corruptorii “Quare” [Glorieux]. 34 In January of 1279, John Pecham, an infl uential Franciscan theologian, was

appointed Archbishop of Canterbury, and, late in the year 1279, William de la Mare completed, possibly under the patronage of Archbishop Pecham, a compilation of errors from the works of St. Thomas. William’s Correctorium Fratris Thomae [Glorieux] includes 117 excerpts from Aquinas’s works.

35 William says, for example: “Ad illud tamen quod multitudo formarum est contra rationem unitatis, dicendum quod res in qua est multitudo et gradus formarum, est una per formam ultimam, ut expresse dicit Philosophus VIII Metaphysicae et Com-mentator . . .,” Correctorium Fratris Thomae [Glorieux], p. 396.

36 For discussion of this Franciscan endorsement of William de la Mare’s Correcto-rium, along with detailed consideration of the response of followers of Aquinas, see Roensch, Early Thomistic School.


theses, the actuality of matter37 and the plurality of substantial forms, supporting thereby TP1–TP5, and it contributed to the entrenchment of these theses, and to the division of Franciscans and Dominicans on philosophical grounds.38 A complex of doctrines, viz., TP1–TP5, along with, for example, the view that matter is not the principle of individua-tion, and the rejection of the Thomistic distinction between essence and existence, remained associated with the distinctive and quite long-lived approach that I have referred to as scholastic pluralism.

Wyclif argues that, from the Thomistic viewpoint, the composition of a human being is as simple as that of an element, which is absurd. Like his Franciscan predecessors, Scotus and Ockham, he maintains that human beings and other compound substances have a pluralistic structure. He claims, for example, that “many substantial forms of different species are together in the same composite, however one will be subordinate to another, as is clear of a compound.” and “one form that is more general and another more specifi c, which are in the same suppositum, are ordered together, as forms of different species, as is manifestly clear of bone, of fl esh, of nerve and other heterogeneous parts in man.”39 So, in Wyclif ’s view, Bossy the cow is composed of a plurality of grades of matter and of substantial forms. First, prime mat-ter is the subject of the substantial forms of the elemental atoms,40 and differing combinations of the elements, i.e., of earth, air, fi re, and

37 Among William’s arguments for his position, as especially important, are argu-ments that focus upon God’s omnipotence. For example: (1) The Thomistic view that matter is pure potentiality, with no positive reality, implies that God cannot make matter without form (Correctorium, p. 409, art. 108).; that God cannot cause matter to precede its form in time (ibid., p. 113, art. 27), and that God can have no idea of matter simpliciter (ibid., p. 326, art. 79; p. 389, art. 97; see also articles 4, 80, 81). But this is a denial of God’s omnipotence, and so must be false. The last point, of course, also questions divine omniscience.

38 Various lengthy replies to William’s Correctorium by Dominican Thomists appeared in short order. Four, entitled the Correctorium corruptorii, have been identifi ed respectively as “Quare,” “Circa,” “Sciendum,” “Questione,” and the fi rst three, with much dis-agreement among scholars, have been respectively attributed to Richard Knapwell, John Quidort, and William of Mackelsfi eld. See Glorieux, Le Correctorium corruptorii “Sciendum,” vol. II, pp. 12–19.

39 Wyclif, Trialogus, p. 92: “. . . multas formas substantiales dispares specie esse in eodem composito dum tamen una sit subordinata alteri, ut patet de mixtis. Unde sciendum, quod sicut una forma generalior et alia specialior, quae sunt in eodem supposito, ad invicem ordinantur, sic formae disparium specierum, ut manifeste patet de osse, de carne, de nervo et caeteris partibus etrogeneis in homine.”

40 Wyclif ’s distinctive account of prime matter and his analysis of elemental atoms, each of which is a composite of prime matter and a substantial form, will be examined in following sections.

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water, in turn, are the matter of the substantial forms of the minima naturalia (the compound corpuscles) that compose Bossy’s blood, bones, fl esh and the like.41 Next, Bossy’s body is the matter of a supervening substantial form, i.e. Bossy’s cow soul. Scholastic pluralism provided an approach to corporeal substances that made possible an atomism that is consistent with a hylomorphic ontology. His pluralistic analysis, therefore, provided the framework for Wyclif ’s atomistic account as well as for his dualistic account of the human body and mind.

2. The Final Question

2.1. The Context of Wyclif ’s Atomism

The fi nal question posed above is:

Q3: Why, in Wyclif ’s view, is a corpuscular matter theory needed in conjunction with the Aristotelian view that prime matter and substantial form are the fundamental principles of corporeal things?

To respond to this question, we turn fi rst to Wyclif ’s context.In the fourteenth century, the nature of prime matter became a

hotly debated philosophical problem. As noted above, Aquinas’s view provided grounds for identifying prime matter with pure potentiality. He, therefore, says that prime matter “does not have being (existence) per se.”42 He explains:

It is sometimes under one form, and sometimes under another. But through itself matter can never exist, because—since in its nature it does not have any form, prime matter does not have existence in actual-ity, since existence in actuality is only from a form. But [prime matter is] only in potentiality. And so whatever is in actuality cannot be called prime matter.43

41 Wyclif used the term “minima naturalia” to refer to specifi c secondary particles, composed of atoms. Pierre Gassendi (1592–1655) is generally credited with the fi rst use of the term “molecule” to identify secondary or compound particles.

42 Thomas Aquinas, De Principiis Naturae [trans. Bobick], p. 31: “Sed per se numquam potest esse.”

43 Ibid.: “Quandoque enim est sub una forma, quandoque sub alia. Sed per se numquam potest esse, quia—cum in ratione sua non habeat aliquam formam, non habet esse in actu, cum esse in actu non sit nisi a forma. Sed solum in potentia. Et ideo quidquid est in actu, non potest dici materia prima.”


Therefore, although it is the persistent substratum, in which all sub-stantial forms inhere and which itself inheres in nothing else, prime matter is pure potentiality.

Scotus, a thoroughgoing scholastic pluralist, sought to make clear inadequacies of this account of prime matter. He distinguishes subjec-tive potentiality from objective potentiality:

For something can be in potentiality in two ways: In one way as an end; in the other way as a subject that is in potentiality toward an end (. . .) so that the subject existing is said in subjective potentiality; and the same in respect to the agent is said to be objective; they can be said to be sepa-rated, as in creation, where it is objective potentiality and not subjective, because here something is not a subject.44

His analysis and arguments raised consciousness in regard to the claim that matter, to serve as a principle of generation and corruption, must have subjective potentiality ( potentia subjectiva).45 That is, prime matter must be an actual subject to receive substantial forms.46 This means, Scotus concludes, that prime matter must actually exist, and, what is

44 John Duns Scotus, Sent., lib. 2, dist. 12, q. 1 [Vivès], p. 556: “Aliquam enim potest dupliciter: Uno modo ut terminus; alio modo ut subjectum, quod est in potentia ad terminum . . . ita quod subjectum existens dicitur in potentia subjectiva; et eadem ut respicit agens, dicitur objectiva; possunt tamen separari, ut in creabili, ubi est potentia objectiva et not subjectiva, quia ibi non subjicitur aliquid.”

45 Ibid., pp. 556–558. The Scotistic notions of subjective potentiality and objective potentiality are clarifi ed by a late Renaissance Scotist, Fillipo Fabri, who explains that according to Scotus: “Something can be in potentiality in two ways, one way as the end of the action of an agent, so that is said to be in potentiality that is in itself nothing, but is in the power (virtute) and the potentiality of an agent, as the world before it was created was in potentiality and in the power (virtute) of the agent God; but in itself it was nothing, and this potentiality he names objective potentiality; another way that something is in potentiality is in respect to a capacity (capax) of a subject and the recep-tion of whatever forms, and this is subjective potentiality.” Cf. Fillipo Fabri, Philosophia naturalis Joan Duns Scoti ex quatuor libris Sententiarum et Quodlibetis collecta [Venice, 1616], p. 126: “. . . aliquid potest esse in potentiae dupliciter, uno modo, ut terminus actionis agens, et sic dicitur esse in potentia illud quod ex se nihil est, sed est in virtute, et potentia agentis, ut mundus ante quam crearetur erat in potentia et virtute agentis Dei; sed in se ipso nihil erat, et hanc potentiam appellat potentiam objectivam, alio modo aliquid est in potentia quatenus est subjectum capax, et receptivum alicuius formae, et haec est potentia subjectiva.”

46 Ibid., p. 558: “Dico igitur, quod materia est per se unum principium naturae, ut dic Philosophus primo Physicorum . . . Quod est per se subjectum mutationum sub-stantialium, quinto Physicorum . . . Quod est terminus creationis; igitur sequitur quod est aliquid, non in potentia objectiva tantum, . . . sed oportet tunc quod sit in potentia subjectiva existens in actu . . .”

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more, must be capable of existing independently, apart from form.47 He further justifi es his contention that prime matter has an “actus” (an act or actuality) of its own. Aristotelians distinguished two fundamen-tal kinds of acts that a substance has, fi rst the act of simply being (or existence) and second those acts that follow from its nature, e.g., fi re moving upwards or earth falling downwards. Substantial form is the principle of these two acts in each substance. Prime matter, though, has just the former act, an entitative act (an act of being), that gives it actuality (actual existence).48

Scotus objects that in Aquinas’s view, prime matter is accorded only “objective potentiality” ( potentia objectiva). That is, prime matter has reality only in regard to some effi cient cause and is merely the poten-tiality to become some x (a future object that does not yet exist), as the universe has objective potentiality in regard to God as its effi cient cause, before it is created by Him. Aquinas’s prime matter, therefore, has no reality in itself (it is, in effect, nothing) and, therefore, cannot serve as the continuous and stable subject of substantial forms required for substantial change.49 The subtle doctor’s arguments had a dramatic effect upon fourteenth century matter theory. The intuition that prime matter must have a robust reality to serve as a material cause in sub-stantial change inspired a variety of subsequent theories.

A further problem was that of making sense of the extension natural to a body composed of prime matter and substantial form, if neither has extension, an issue raised by Averroes. His claim, in his De substantia orbis,50 that prime matter has indeterminate dimensions, in the light of Scotus’s opposition to all attribution of quantity to prime matter, focused attention upon this issue. The intuition that the extension of bodies must have its foundation in the nature of the matter of a sub-stance inspired a variety of fourteenth-century interpretations of this Averroist view.

47 John Duns Scotus, Sent., lib.2, dist. 12, q. 2, p. 576: “Sed forma est causa secunda, quae non est de essentia materiae . . .; ergo Deus sine illa potest facere materiam.”

48 John Duns Scotus, Ibid., p. 589: “. . . et sicut forma est actus formalis, quia potest informare per receptionem ipsius, ita etiam materia est actus entitativus et positivus vere receptivus ipsius formae, et nihil amplius habens.”

49 John Duns Scotus, Sent., lib. 2, dist. 12, q. 1, pp. 559–561. See also discussion of Scotus’s notions of subjective potentiality and objective potentiality by Cross, The Physics of Duns Scotus: The Scientifi c Context of a Theological Vision, pp. 17–20; and McCord Adams, William Ockham, pp. 642–643.

50 Averroes, Sermo de substantia orbis [Venice, 1562], p. 4: “. . . hoc subjectum recipit primus dimensiones interminates.”


We fi nd here stimuli that inspired two hotly debated fourteenth century issues in matter theory:

MQ1: Does prime matter have an actuality [distinct reality] of its own (as Scotus claims)?

MQ2: How does the extension of bodies have its foundation in prime matter?

Ockham provided a distinctive response to these queries. Like Scotus, Ockham supports the dictum: prime matter has an actuality of its own. Ockham says that “matter is a certain thing actually existing in the nature of things, which is in potentiality towards all substantial forms;”51 but this potentiality is unlike “the way in which a future white is just in potentiality.”52 This potentiality is not simply the absence or a lack of some objective state, i.e., the non-existence [or privation] of a future white which can be said to be in potentiality (this lack or absence is like Scotus’s objective potentiality). Instead, prime matter is a thing that is in potentiality towards a form that it lacks (as Socrates now tan is potentially white).

Further, Ockham maintains that “potentiality is not some existing thing in matter, but it is matter itself,”53 because this potentiality does not cease when matter receives a form. Matter is potentiality in respect to substantial forms generally. It may be in potentiality to one particular form or other, but even when this form is received, the potentiality in respect to substantial forms generally is not corrupted. The prime matter of air is still able to be [posse] air, but it is also potentially fi re or water. Hence, potentiality is not merely a property of matter. Instead:

O1: Prime matter is potentiality.

So Ockham says that potentiality, and so prime matter, remains along with form. Further, prime matter can neither be generated nor

51 William of Ockham, Summula philosophiae naturalis [Brown], p. 179: “Materia est quaedam res actualiter exsistens in rerum natura, quae est in potentia ad omnes formas substantiales.”

52 Ibid., p. 179: “. . . modum quo albedo futura est tantum in potentia”. 53 Ibid., p. 185: “. . . potentia non est aliqua res existens in materia, sed est ipsa

materia.”

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destroyed, so it has no potentiality to not exist. It therefore must be something that is “truly actual from itself.”54 He also argues:

That which is not can be the part or principle of no being; but matter actually is a part and a principle of a composite being. Therefore, it itself is actually a being in actu [act or actuality].55

So prime matter actually exists; it “is truly a being [ens] in actu.”56 Ockham concludes:

O2: Prime matter is some thing [res] actually existing and distinguish-able from form.

Further, Ockham makes his prime matter a yet more robust “being in actuality.” Claiming that each entity has its individual prime matter (I have my prime matter and you have yours), he also supports the following view:57

[I]t is impossible that matter should lack extension, for it is not possible that matter should exist without having one part at a distance from another part. (. . .). [T]he parts of matter can never be in the same place. There-fore, matter always has one part at a distance from another part.

The Venerable Inceptor here argues that for matter to be divisible into parts, extension (length, breadth, and depth) is essential to matter. Distinctive of Ockham’s account of matter is his claim:

O3: Quantity or extension is not anything apart from prime matter.

Prime matter essentially has quantity or extension, which is here ana-lyzed as parts outside of parts. Matter is not different from its parts, nor is it therefore different from extension or quantity or dimensions.58

54 Ibid., p. 180: “. . . vere actu ex se ipsa”.55 Ibid., p. 186: “. . . illud quod non est, nullius entis potest esse pars vel principium;

sed materia actualiter est pars et principium entis compositi, igitur ipsa est actualiter entitas in actu.”

56 Ibid.: “. . . sit vere ens in actu.”57 Ibid., p. 191: “. . . impossibile est quod sit materia sine extensione: non enim est

possibile quod materia sit nisi habeat partem distantem a parte . . . numquam partes materiae possunt esse in eodem loco. Et ideo semper materia habet partem distantem a parte.”

58 Ibid., pp. 191–192.


Wyclif fi nds unsatisfactory Ockham’s response to these two questions. In regard to MQ1: Is prime matter an actual entity?, Wyclif agrees with Ockham’s contention (O2) that prime matter has a positive reality of its own. But how can a potentiality have actuality? If, as Ockham claims, prime matter is potentiality simpliciter (O1), how can prime mat-ter have a positive reality? Still, Wyclif says (Tr, p. 87), citing Aristotle, that prime matter has no quiddity, no quality, no character at all.59 He takes this claim very literally. This raises the problem: How can that which has no character whatsoever have a positive reality?

In regard to MQ2: in which way does extension (or quantity) have its source in prime matter? Wyclif claims, as noted above, that prime matter has no quantity or quality. Prime matter, therefore, cannot essen-tially be identifi ed with extension, as Ockham claims (O3). But Wyclif, nonetheless agrees that prime matter is the source of extension. This raises the perplexing question: How can prime matter, independently of any quantity or characteristics in itself, be the source of extension?

Interestingly, Wyclif ’s distinctive atomism contributes to his response to these quandaries. His atomism, therefore, is not only consistent with his hylomorphism (scholastic pluralism), i.e., with his scholastic matter/form theory. His atomism enables him to resolve problems in matter theory that, in his view, confounded his Aristotelian predecessors and contemporaries.

We turn fi rst to MQ1 (Is prime matter is an actual entity?). In his Logicae continuatio, Wyclif provides his prime matter with a palpable reality. He contends that prime matter is:

composed from indivisibilia [indivisible points], and occupying every pos-sible place, not corruptible according to any of its parts, except acciden-tally by the division or separation of one of its parts from the rest.60

Likewise, he explains, in his Trialogus that time is a continuum composed of indivisibilia, i.e., instants, and so too prime matter (“occupying every possible place” in the world) is a continuum composed of indestructible indivisibilia (indivisible points). [Tr, p. 88] Although each indivisible point has no quantity, no quality, no character at all, indivisibilia make his prime matter entitative. Hence, prime matter has actuality.

59 Wyclif, Trialogus [Lechler], p. 87: “Aristoteles autem dicit septimo Metaphisicae, quod nec est quid nec quale nec aliquid aliorum entium . . .”

60 LC3.9, p. 119: “. . . esse ex indivisibilibus composita, et occupare omnem locum possibilem nec esse secundum eius partem aliquam corruptibilem, nisi forte per divi-sionem vel separacionem unius partis a reliqua.”

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In response to MQ2 (In which way does matter have quantity?), Wyclif ’s contends that indivisibilia, constituting prime matter, also provide it with extension. He claims: “the world is composed from certain atoms, and cannot be increased or diminished, or arranged in a straight line or another fi gure, for the number of atoms [multitudinem athomorum] is a cause from which such a continuous quantity and such a fi gure naturally immutably follows.”61

In his view, the totality of indivisible points or atoms defi nes the matter of the world. Further, each indivisible point has its position relative to fi xed points at the centre and poles of the world, and no two points can occupy the same position. As a result, indivisibilia, though unextended, are, in effect, impenetrable, so they cannot overlap or interpenetrate each other, nor can there be an actual place where there is no point (because every line is actually composed of points). As a result, indivisibilia must be located contiguously, one next to the other. This entails, in Wyclif ’s view, that the totality of contiguous corporeal indivisibilia defi ne the total size and shape of the world, and, thereby, its extension. Although Wyclif rejects Ockham’s view that extension belongs to prime matter, extension is produced by the contiguous indi-visibilia that constitute prime matter.62

Wyclif, therefore, responds to these fourteenth-century concerns by employing indivisible points (identifi ed as primitive atoms) both to provide matter with a robust reality (actuality) and to account for how matter provides extension to corporeal things. We might, however, note that though Wyclif ’s atomistic account is effective in resolving some problems in matter theory, this utility of his indivisibilia does not entail the existence of atoms. One might argue that this provides some rationale for accepting indivisibilia. But it does not explain the grounds and motivation for Wyclif ’s atomism. We are still left with the question of why Wyclif adopted an atomistic account, especially in the light of the almost universal rejection of atomism and arguments against it provided by his contemporaries.

61 LC3.9, p. 1: “. . . mundum componi ex certis athomis, et nec posse majorari nec minorari nec moveri recte localiter vel aliter fi gurari, ita quod tantam multitudinem athomorum consequitur tanta quantitas continua et talis fi gura, propter causas immu-tabiles naturales.”

62 It should be noted that, although Wyclif adopts an atomistic account, he clearly departs from the Ancient atomists in his rejection of a void.


Second, we should also note that Wyclif goes beyond indivisibilia theory. He develops a full blown atomistic account that presumes elemental corpuscles and, in turn, compound particles. This also raises the question of the reason for his commitment to an atomistic account. Following is a further exploration of this question. What I wish to show is that his atomism was motivated and inspired by his distinctive meth-odology, which he identifi es as the “logic of scripture” (logica scripture). Although this too will not provide us with a proof that atoms exist, it will provide further insight into the rationale for Wyclif ’s atomism.

2.2. Why did Wyclif Develop a Corpuscular Matter Theory? Wyclif ’s Methodology

Wyclif adopted a distinctive methodology for the acquisition of truth. In his De Veritate Sacrae Scripturae, he contends that all truth is in Scrip-ture.63 As such, we turn to Scripture as the source of all truths. This includes truths of natural philosophy. Further, Wyclif contends that, though Scripture provides only truth, the language of Scripture is frequently fi gurative. This requires appropriate interpretation. We are aided in this endeavour, in particular, by what Wyclif calls the logic of scripture,64 which he explores and expounds in his lengthy Tractatus de logica. From this viewpoint, logic, properly employed, follows and clari-fi es Scripture. In turn, the logic of Scripture enables us to understand how to retain consistency of interpretation throughout Scripture, to analyze conceptions, including those involving equivocation, to identify the various well-formed structures of terms and correctly interpret these, to draw valid consequences from inter-connected propositions, to formulate sound arguments in support of interpretations of Scripture, and, thereby, to make correct judgements in interpreting Scripture and in supporting interpretations of Scripture.

Intensifying Wyclif ’s confi dence in his view is his Augustinian com-mitment to the foundation of truth in faith, which he claims is cor-rectly given only by Scripture. Further, reason supports what we here are taught by faith. Reason plays a signifi cant role for Wyclif. He takes seriously the Augustinian claim that reason enables us to understand

63 De Veritate Sacrae Scripturae [Buddensieg], hereafter cited as DVSS. For an excellent abridged translation, see John Wyclif, On the Truth of the Holy Scripture [trans. Levy].

64 Wyclif explains (Tractatus De Logica, Proemium): “Motus sum per quosdam legis dei amicos certum tractatum ad declarandam logicam sacre scripture compilare.”

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what we believe. Through the logic of scripture, reason provides clarifi -cation and confi rmation of what we believe by faith. Therefore Wyclif presumes the following methodological principle:

The authoritative source of all truth is scripture. The correct inter-pretation of scripture is guided by the logic of scripture.65

His logic of scripture would seem to have its foundation in a method for the acquisition of all knowledge, natural and spiritual, much like the following:

1. Guided by logic, i.e., an understanding of literal and fi gurative lan-guage, the logical role of terms and of propositions, and the like: formulate an interpretive thesis of a scriptural truth.

2. Use logic (rules of the relations of propositions, rules of infer-ence, etc.): to test for internal consistency with other propositions of scripture; to test for support from interpretations provided by reliable commentators, e.g., Wyclif mentions Augustine, Jerome, Ambrose, Lincoln,; to test for consistency with evidence of sense and reason.

3. Use logic to: draw conclusions from the thesis presumed.4. Use logic to: test conclusions for consistency with other scriptural

truths; with evidence of sense and reason.5. If inconsistencies (contradictions) result, reject the interpretive thesis.

If it is consistent and well supported, retain the thesis.

In Wyclif ’s view, his logic of Scripture provides the principles of cor-rect reasoning that aid us in interpreting scripture accurately and in understanding what we believe by faith. Scripture, thereby, provides the “matter of knowledge” (materia de scire)66 and is the focus of his discussion of logical proofs.

Our Oxford master, therefore, explains in his discussion of prime matter that “ignorance of true physics makes moderns give too little weight to Scripture and to impose upon the sacred doctors, ignorance of logic or philosophy . . .”.67 Instead, “Scripture will be the model . . .,

65 Wyclif, DVSS, esp. part 1, ch. 3.66 Wyclif, Tractatus de logica, Proemium.67 Wyclif, De M&F, p. 218: “Ignorancia huiusmodi veritatum physicarum facit

modernos parum pendere scripturam et imponere sanctis doctoribus ignoranciam loyce vel philosophie . . .”


not only in respect to right living, but also in speaking the truth and philosophical wisdom. Now however . . . we make our logic the rule cor-recting scripture, . . . when, however, it ought to be on the contrary.”68 From this viewpoint, there are no foolproof authorities, but some early doctors of the Church, especially Augustine, are a more reliable guide to truth because they are so well attuned to the meaning of scripture. So, in response to some objections, Wyclif says: “For a solution to these, one ought to assume the grammar and logic of the holy doctors, which they elicit from Scripture.”69 He also tells us that other or secular (in Wyclif ’s words, ‘foreign’) logics are numerous (as many as there are logic masters) and shortlived (“a foreign logic endures for barely twenty years”).70 “The logic of Scripture, however, stands eternally, because it has been established by the indestructible truth . . .”.71 Nonetheless, Wyclif contends that “Aristotle’s logic is correct for the most part, and consistent with the logic of Scripture.”72 Still, Aristotle, though a great philosopher, has committed errors. [DVSS, p. 29] Aristotle correctly understood the logical investigation of language, but logic cannot be secure independently of a foundation of truth that is its subject, and this foundation of truth is the words of Scripture.

Using this method, Wyclif explains the account of prime matter that he learned from Scripture by interpreting the “true proposition, Genesis 1: ‘In the beginning God created heaven and earth’.”

“In the beginning God created heaven and earth.” That is: In the word God created a spiritual and a corporeal creature. From which, ordered also with admirable subtlety common to all, he calls the same creature the said corporeal essence (under the idea that is matter) earth, water, and abyss, because common people cannot understand the corporeal nature under the idea that is matter; so it is necessary to have these express the names of sensible things, that are most unformed. Nor is it falsely named, as is said after; and towards showing that it is unformed, he calls it void, vacuum, and darkness. These said privations however are nothing except

68 Ibid., p. 219: “. . . scriptura fuit exemplar . . ., nedum ad recte vivendum, sed ad vere loquendum et philosophice sapiendum. Nunc autem . . . constituimus loycam nostram tanquam regulam rectifi cantem scripturam, . . . cum tamen debet esse e contra.”

69 Ibid., p. 235: “Pro solucione istorum oportet supponere grammaticam et loycam sanctorum doctorum, quam eliciunt ex scriptura.”

70 DVSS, p. 54: “. . . vix durat una aliena logica per viginti annos . . .”71 Ibid.: “Logica autem scripture in eternum stat, cum fundatur independenter a

fama vel favore hominum infringibili veritate.” 72 DVSS, p. 47: “. . . logica Aristotelia, que ut plurimum est recta, sit logica scrip-

ture . . .”

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this unformed matter. After this (nature not time) is, in fact, the order of the fi rst day, light, in substantial form in the fi rst instant of time and matter naturally, not temporally, before that same instant.73

Wyclif formulates an interpretive thesis of the scriptural truth provided in Genesis 1, namely: “In the word,” i.e., in His act of creation, God created two kinds of creature, corporeal and spiritual. And, alterna-tively, he says: In the beginning, “the fi rst essence (i.e., God) before all substance created two essences, . . . namely, a corporeal essence and an incorporeal essence.”74 Further, looking at Genesis 1 beyond this pas-sage (“earth, water, abyss,” “void, vacuum, darkness”), and considering Augustine’s exegesis,75 Wyclif analyses the latter (essencia incorporea) as angels (immaterial substantial forms that exist per se) and the former (essencia corporea) as unformed prime matter (matter without any inhering material substantial form). He further explains this passage.

Wyclif describes the “incorporeal essence” as “pure per se stans,”76 “a substance that is indivisible in respect to mass and intelligible in respect to operation, which are creatures most proximate to God, which philosophers in respect to their innate action call intelligences, and in respect to their duty to us they call these angels.”77 The incorporeal essence consists in separately existing individual created Minds, intel-ligibilia, created immediately by God, with powers of intellect, will, and memory. Wyclif adopted the common scholastic view that each angelic substance is indivisible, because what is immaterial is not extended, and only extended entities are divisible.

73 De M&F, pp. 209–210: “In principio creavit deus celum et terram; id est: In verbo creavit deus spirtualem creaturam et corpoream. Unde ordinate et mirabili subtilitate communiter eandem creaturam vocat dictam essenciam corpoream (sub racione qua est materia) terram, aquam et abissum, quia rudis populus non suffi ciebat comprehendere naturam corpoream sub racione qua materia; ideo necesse habuit illam exprimere nominibus rerum sensibilium, que maxime accidunt ad informitatem. Nec false nomi-nat, ut post dicetur;. et ad testandum eius informitatem dicit eam inanem, vacuam et tenebrosam. Dicte autem privaciones non sunt nisi informitates materie huius. Post hoc (natura non tempore) facta est, prima die ordinis, lux in forma substantiali in primo instanti temporis et materia naturaliter, non temporaliter, ante idem instans.”

74 DCH, p. 24: “. . . prima essencia ante omnem substanciam creavit duas essencias, . . ., scilicet essenciam corpoream et essenciam incorpoream.”

75 De M&F, p. 209: “Et ille est sensus Augustini, 12 de Confessione, ubi, priusquam declarat angelum et materiam priman esse creaturam dei . . .”

76 De M&F, p. 177.77 Ibid.: “. . . substancie quoad molem indivisibiles et intelligibiles quoad operacionem,

que sunt creature deo proxime, quas philosophi ab innata accione vocant intelligencias, et nostri ab offi cio vocant eos angelos.”


Further, bodies are composed of prime matter and material substan-tial forms. In Wyclif ’s view, unlike spiritual substantial forms, which exist per se and can never inhere in prime matter, material substantial forms inhere in prime matter and cannot exist apart from prime matter.78

Prime matter, however, has a robust reality. First, prime matter is the corporeal essence. That is, this essence (prime matter) provides the material foundation that makes my cat Jake, my cherry tree, my gold ring each a corporeal thing. Further, as noted above, Wyclif describes prime matter as “composed from indivisibilia” [LC 3.9, p. 119]. And, likewise, in his Trialogus, he says: “componitur ex suis partibus quantitativis usque ad sua indivisibilia . . .” [Tr, 88].

As such, Wyclif ’s interpretation of Genesis 1 has the consequence that: In the beginning, God created two kinds of indivisibilia, spiritual and corporeal. So, Wyclif explaining the grades of simplicity in the created world, tells us, fi rst, that the highest grade of simplicity is “the divine nature,”79 which, existing in eternity (outside of the instants of time and all other created indivisibilia) excludes all possibility of variation, succession, or of composition from parts. “In the second grade,” i.e., the fi rst grade of simplicity of the created world, “are created spirits and other indivisibilia without quantitative parts, such as points, instants, and the like.”80 The latter are the fundamental parts of material things, i.e., of prime matter, from which, along with substantial form, are com-posed elemental corpuscles, and these, in turn, combine to form compound particles (minima naturalia), which are identifi ed by Wyclif with subsequent higher grades of simplicity respectively.81 These grades of matter will be explored in greater depth in the following section.

78 Tr, p. 87: “Non autem intelligo formam substantialem materiae primae esse aliquid, quod potest per se existere . . .”

79 De M&F, p. 199: “In summo igitur gradu simplicitatis est natura divina, exclu-dens possibilitatem ad quamcunque composicionem ex paratibus, vel variacionem in accidentibus . . .”

80 Ibid.: “In secundo gradu sunt spiritus creati et alia indivisibilia quoad partes quantitativus, ut punctis, instans, et similia.”

81 Ibid.: “In tercio gradu sunt materia et forme substanciales vel accidentales, non habentes partes quantitativas disparium naturarum.” (matter and form considered separately). “In quarto gradu sunt quatuor elementa, quorum quelibet pars quan-titativa est eiusdem nature cum toto.” (elemental corpuscles, and collections of the same kind of elemental corpuscle); and “in isto gradu sunt multa omogenia, quorum quelibet pars quantitativa per se sensibiliter est eiusdem nature cum suo toto, ut caro, os, nervus, et cetera” (compound particles, and collections of the same kind of com-pound particle).

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Wyclif ’s atomism further explains the account of creation in Genesis 1 and is consistent with his interpretation of Scripture more generally. Further, his indivisibilia help him to resolve a variety of objections that features of his interpretation can raise.82 Consideration of this role of Wyclif ’s indivisibilia is a complex project and beyond the scope of this chapter. We might just note that the consistency with a variety of pas-sages in scripture and the explanatory role of indivisibilia in relation to scripture is important to Wyclif. In addition, he cites Scripture in direct support of indivisibilia.

In his Trialogus, Wyclif raises the question of whether a body or whatever continuum is composed from parts that are indivisible and not extended, or divisible and extended.83 If the second, he says, then all parts are always further divisible into other parts, so the number of parts assigned to A in the beginning is not the total number of parts in A.84 He responds, citing Gen 1:31 (“God sees all that He made.”)85 in defense of indivisibilia: “I suppose with Augustine and the faith of Scripture, that just as God sees all that he made, so he distinctly under-stands all parts of whatever continuum, so that no further or different components of a continuum can be given. And on that basis the rea-soning seems plainly to proceed.”86 Wyclif here supports the view that

82 See, for example, his discussion (De M&F, pp. 187–88) of how an explanation consistent with nature can be provided for such seeming miracles as the change of Lot’s wife to a pillar of salt by understanding this as a rapid relocation, possible for God, of salt particles from other parts of her body to her surface. Likewise, explaining that the change of water to wine is not the creation of a new substance (De M&F, p. 188): “Facillimum namque est auctori nature capere minucias elementorum vel inordinate sparsas vel noviter generatas, et armonice componere illas, ut forma serpentis vel vini, vel quecunque alia de potencia materie educibilis, statim resultat; cum nichil ibi creatur, sed vel generatur pure naturaliter, vel prius generatur aliter situaliter.”

83 Wyclif, Tr, p. 83. “Capiatur A corpus vel quodcunque continuum, et noto omnes ejus partes, ex quibus componitur; et quaero ulterius, utrum sunt indivisibiles et non quantae, vel divisibiles atque magnae.”

84 Ibid.: “Si secundo modo, tunc omnes illae partes et quaelibet earum est divisibilis in partes ulteriores, ergo numerus illarum assignatus in principio non est numerus totalis omnium partium A signati.”

85 See also Kenny, Wyclif, p. 62: “But the work frequently repeats that scripture con-tains all truth, and in one passage Wyclif even offers to prove his own atomic theory out of it, from the beginning of Genesis and from Matthew 10:30.” This citation from Matthew is as follows: “But the very hairs of your head are all numbered.”

86 Wyclif, Tr, p. 83: “In ista autem responsione suppono cum Augustino et fi de scrip-turae, quod sicut Deus vidit cuncta quae fecerat, sic distinctissime intelligit omnes partes cujuscunque continui, sic quod non est dare ulteriores vel alias ipsum continuum componentes; et sic videtur ratio plane procedere.”


God’s distinct understanding of “all parts of whatever continuum”87 entails a fi xed total number of primitive parts that are not further divisible (“no further or different components of the continuum can be given”).88 This interpretation of Scripture led him to the twofold consequence of supporting fi nitism and atomism. Wyclif contends that only God, existing in eternity (outside of the instants of time and points of space), is infi nite. He says: “It is impossible that any substance or points or any other thing besides God is simply infi nite.”89 Wyclif, therefore, asserts the following principle of fi nitude: whatever actually exists is fi nite.90 Consistent with this principle, he concludes that every continuum, including the created universe as a whole, must have mini-mal parts that cannot be further divided. This entails that prime matter cannot be infi nitely divisible. Instead prime matter is, in Wyclif ’s words, composed from indivisible points or atoms [Tr, p. 88].

He further supports this claim by citing Ecclesiastes 1891 and 192 which he interprets to mean fi rst, that God created a complete world, every instant of time and whatever exists in time, however simple or complex, each at its proper time and place.93 What is more, this entails that God eternally has a distinct Idea of every part of His Creation.94 S.H. Thom-son points out, as a foundation for Wyclif ’s atomism, Wyclif ’s identifi ca-tion, in De Materia et Forma95 of the possible with God’s knowledge of his creation (that is, whatever has real possibility is in God’s mind because,

87 See preceding footnote.88 See preceding footnote.89 See, for example, LC 3.9, p. 36: “Unde impossibile est quod aliquis numerus

substanciarum vel punctorum, vel aliud preter deum sit simpliciter infi nitum.”90 For Wyclif ’s support for fi nitism, see also LC 3.9, pp. 37–38. Wyclif contends (LC

3.9, 37): “Omnem ergo numerum qui excedit ingenium nostrum ad aptandum sibi terminum specifi cum naturalem vocamus infi nitum.”

91 De M&F, p. 222: “Qui vivit in eternum creavit omnia simul.”92 Ibid.: “. . . nihil novi sub sole.”93 See, also Wyclif ’s claim that, at the beginning of time, God created all things

simultaneously, Tr. p. 86: “Deus creavit omnia simul;” “. . . qui vivit in aeternum, creavit omnia simul,” and also LC 3.9, p. 56: “. . . Deus ordinat istos propter melius ordinis universi.”

94 See, for example, LC 3.9, p. 36: “Ymmo deus satis noscit quomodo omne quadra-tum per se sensible integratur ex partibus minimis et principiis eorum indivisibilibus cumulatis. Et sic dicitur de qualibet alia fi gura principiata ab indivisibilibus primi numeri, ut fi gurati. Novit eciam in qua proporcione quicunque numerus punctorum se habet ad alium . . .”

95 De M&F, pp. 234–235.

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as Wyclif says: “God sees all that he has made.”).96 Thomson, explain-ing the rationale for Wyclif ’s atomism, says: “Increase or decrease in the magnitude of the universe is likewise unthinkable, for whatever has being already exists in God’s thought . . . That is to say, that whatever is known to God has reality, and the smallest possible unit of space or time, the point or the instant becomes quite real.”97

Kretzmann similarly says: “Why, then, was Wyclif an indivisibi-list? . . . All that we have seen that might have prompted such a belief on his part is certain theological considerations, especially concerning the requirements of omniscience, and I think there can be no doubt that they are indeed the bedrock of his indivisibilism.”98 He explains this claim:

God alone knows the detailed composition of things out of indivisibles, but in Wyclif ’s view that sort of knowledge necessarily includes knowing the precise number of the indivisible constituents of the world and of each thing in it. If there are literally infi nitely many points in a line, then, it is logically impossible that anyone, even omniscient God himself, can know the number of its points. And the principle reason why it will not do to say simply that God knows that there are infi nitely many of them is, I think, that since they are real and natural, God made them; and omniscient God must know each of his creatures individually. As Wyclif puts it more than once, quoting or paraphrasing Genesis 1:31, ‘God sees all the things that he has made.’99

Like Democritus and Epicurus, Wyclif claims that his primitive atoms (i.e. points) are indivisible, but, quite unlike Democritus and Epicurus, interestingly enough, Wyclif ’s fundamental motivation for this view is theological. It is his interpretation of Scripture, in particular, that inspired Wyclif ’s atomism. As required by his methodology, this inter-pretation of Scripture is further supported by reason and experience. These supporting arguments and the atomistic view that he developed will be considered in the fi nal section.

96 Wyclif says, at this point in De M&F (Ibid.): “Et talis veritas est racio vel exemplar quod deus necessario videt, et videndo illud quod est essencialiter divina essencia, videt omnes creaturas, si sint in tempore suo.”

97 Thomson, “The Philosophical Basis of Wyclif ’s Theology,” p. 112.98 Kretzmann, “Continua, Indivisibles, and Change in Wyclif ’s Logic of Scripture,”

p. 50.99 Kretzmann, Ibid., p. 45. See Kretzmann (Ibid., pp. 51–63) for Wyclif ’s interesting

discussion of the terms “incipit” and “desinit,” and, in this context, Wyclif ’s indivisibilist analysis of motion. Kretzmann discusses, in particular, LC 2.14 (a chapter in Wyclif ’s Logicae Continuatio that examines the terms “incipit” and “desinit”), pp. 191–195.


3. Wyclif’s Atomism

While their hylomorphic ontology led Wyclif ’s predecessors to reject atomism, he used his hierarchical and pluralistic analysis of composite substances to justify his atomism. From Wyclif ’s viewpoint:

1. Prime matter is constituted by a fi nite number of indivisible unex-tended atoms (indivisibilia).

2. Indivisibilia, united by an appropriate elemental substantial form, are extended elemental atoms of earth, air, fi re, and water.

3. Elemental atoms, in different proportional arrangements, compose minima naturalia (compound particles), which are the fundamen-tal particles of compound bodies. Each compound particle has a supervening substantial form.

4. Homœomerous compound bodies, e.g. blood, bones, fl esh, nerves (each with its distinctive kind of compound particle), compose animal bodies, which are animated by a supervening organic soul.

5. An appropriate animal body united with a spiritual human soul, a human mind, composes a human person [a rational animal].

6. The total set of indivisible atoms, which are the minimal units composing all individual material substances, defi ne the shape and size of the world.

Wyclif cites Democritus and Plato as authorities in support of his atomistic view. [Tr, pp. 83–84; LC 3.9, P. 132].100 He, like Democritus, contends that the fundamental building blocks of natural things are atoms. But, unlike Democritus, Wyclif claims that there are several grades of atoms, all of which were formed by God at Creation. The simplest atoms, like the atoms of Democritus, are indivisible, impen-etrable, immutable, indestructible. But unlike the atoms of Democritus, Wyclif ’s indivisible atoms are unextended. Further, in Wyclif ’s view, extended elemental corpuscles (which, though relatively stable, are divis-ible) are composed of a structure of these unextended indivisible atoms and an elemental substantial form. Next, compound particles (called minima naturalia), in turn, are composed of a structure of elemental corpuscles and a supervening substantial form. Compound particles

100 See also LC 3.9, 61, where Wyclif speaks of the ancient view of fi ve simple corpo-real fi gures associated with the four simple elemental bodies and a neutral element.

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themselves are substances, that, like living things, have a hierarchical and pluralistic structure. Wyclif ’s grades of atoms (i.e., his indivisible points, elemental corpuscles, and compound particles) are the subject of what follows.

3.1. Wyclif ’s Indivisible Atoms

Like in other works of his time, in Wyclif ’s Logicae Continuatio the term ‘indivisibilia’ refers to points of a continuum, in particular, of space, time, or motion. Indivisibilia are unextended entities, and are therefore not physical atoms. But, Wyclif also argues, unlike his contemporaries that bodies are in fact composed of physical atoms, which are themselves composed of indivisible and unextended points (i.e., of indivisible and unextended atoms).101 This requires explanation.

The common view at Wyclif ’s time was the Aristotelian view that a continuum is not composed of actual indivisibilia, but rather is infi -nitely divisible. Wyclif sided, instead, with those who maintained that a continuum is composed of actual points.102 His chief adversaries were the nominalists, the most prominent of whom was medieval giant, Wil-liam of Ockham, who maintained that the term “indivisibilia,” refers to nothing actual; these “beings of reason” have a negative connotation, namely, a lack of extension, like, for example, the limit of a line.103 The most common view of the opposition was that points are merely potential, not actual.

Wyclif ’s divisibilist opponents maintained:

TD1: There are no actual indivisibilia.TD2: Indivisible and unextended points cannot be arranged contigu-

ously to produce a continuous line.

Wyclif maintains instead:

TI1: A fi nite and fi xed number of actual indivisible points determine the extension and fi gure of the universe.

101 See, for example, LC 3.9, 1.102 For an excellent discussion of the status of these views at Wyclif ’s time, see

Kretzmann, “Continua, Indivisibles, and Change in Wyclif ’s Logic of Scripture.”103 William of Ockham, De sacramento altaris [Birch]. For an excellent discussion of

Ockham’s view, see McCord Adams, William Ockham, pp. 201–212.


TI2: Space, time, and motion are all composed of parts or points.TI3: Indivisible points are the prime matter of all bodies in the world.

The evangelical doctor argues that, as Aristotle claims, points are naturally prior to a line, and so are a necessary condition for a line to be. Therefore, points cause a line. But they are not an extrinsic cause. They are therefore an intrinsic cause and are consequently part of a line. [LC 3.9, p. 30] He contends that Aristotle maintains that points compose a line, lines a surface, and surfaces a volume. Further, parts always precede the whole of thing. So points must be actual, as the component parts that compose a line. In addition, if all the points are removed from a line, surely nothing remains; so a line must be com-posed of actual points.

He also argues, citing Aristotle, that an instant is the principle of time and unity of number and each is required for these to be. Like-wise, a point is the principle of a line and therefore actual points are required for a line to be. Otherwise the points contained in a line could be successively removed without reducing the length of the line. So all points could be removed and it could remain the same line. But that which can be removed without changing a subject is an accident of that subject. So that which is the principle of a line will be an accident of that line, which is absurd [LC 3.9, p. 30].

Wyclif cites Lincoln, i.e. Grosseteste, in support of actual points in a line,104 but also disagrees with this predecessor, who claims a line is composed of an infi nite number of actual points. The evangeli-cal doctor, who rejects the possibility of any actual infi nite in nature, agrees instead, for example, with Chatton105 and Crathorn,106 that a continuum must be composed of a fi nite number of points, which are immediately contiguous to one another. He argues: that which is infi -nite cannot be known. But we know continua of space, motion, and time, so the component smallest parts of these, though they cannot be apprehended by our senses are indirectly known by us, and, further, they must actually be known by God.

104 LC 3.9, p. 35.105 For discussion of Walter Chatton, see V. Zubov, “Walter Catton, Gérard d’Odon

et Nicolas Bonet.”106 For discussion of Crathorn’s view, see Robert, “William Crathorn’s Mereotopo-

logical Atomism” in this volume.

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This leads Wyclif to a distinctive account of space or place and of the matter that composes what exists. Points, visible to God, are the component parts of space and the matter of what exists. [LC 3.9, p. 2ff.]107 Space is the locus of points, and the place of a particular point is its site relative to certain fi xed points, those at the centre and poles. Wyclif identifi es the contiguity of two points, A and B, with the absence of anything between A and B. Two points compose a line, three a sur-face, four a volume. In his view, a fi nite number of points defi nes the extension of space,108 and, in accordance with God’s design at Creation, fi xed numbers of points, united by an appropriate elemental substantial form, compose the four elemental atoms, those of earth, air, fi re and water, which, in turn, are the matter of all compound bodies.

Clearly Wyclif ’s indivisible atoms are not like those of Democritus, whose atoms have size, shape, and motion. Wyclif ’s indivisibilia or points have no extension, no qualities, no characteristics at all. Nonetheless, their contiguity produces the size and shape of the world and of the elementary corpuscles that compose that world. Further, Wyclif ’s world is a plenum, so he, also, rejects Democritean motion of atoms in a void. Still, his primitive atoms can move relative to fi xed points at the center and the poles by one point successively replacing another.

3.2. Wyclif ’s Elemental Atoms and Minima Naturalia

In the second part of section one above, we found two confl icting analyses of the ontological structure of material substances, that is, two

107 See Wyclif ’s interesting argument, LC 3.9, pp. 33–34, supporting his view that the world is composed of contiguous points. He argues: God has the power to make a substance the size of a point (“nullus theologus negaret quin Deus de potentia aboluta potest facere substanciam punctualem”), and that if God can do this, he can likewise juxtapose points contiguously (“Nec dubium quin, si Deus potest unum punct[u]ale producere, potest et quodlibet juxtaponere.”). It is clear that, from such points, God can make a single extended substance (“Et ultra patet quod Deus potest ex talibus non quantis facere unum quantum)”. Suppose God creates at every point in the world such a punctual substance, and annihilates all continuous substance, preserving punctual substances. (“Creet Deus ad omnem situm punct[u]alem mundi unam sub-stanciam punct[u]alem, et annichilet post omnem substanciam continuam, servando punct[u]ales substancias inmotas”). Wyclif proceeds to argue that this would change nothing. Further, this is possible. So, it cannot be concluded that this is not so in fact. (“Nec dubito quin, admisso hoc pro possibili, omnes philosophi mundi non haberent infallibilem evidenciam ad concludendum quod non est sic de facto”).

108 See LC 3.9, p. 42 for Wyclif ’s argument that because the number of points composing the world entail its size and shape, so it cannot be larger or other than it is, this nonetheless does not detract from God’s infi nite power.


renditions of how a substance is a composite of prime matter and substantial form, namely, scholastic monism and scholastic pluralism. Scholastics were also divided in their analyses of the physical structure of compound substances. Wyclif here examines the vexing scholastic problem of how, in the generation of a new substance, the elements remain in a mixed (i.e., compound) body.109 He distinguishes three basic sorts of analyses, identifi ed by Wyclif as those of the moderns, of Averroes, and of Avicenna [LC 3.9, pp. 75–79].

Wyclif cites Aristotle’s defi nition of a “mixtio” (compound): “mixtio est miscibilium alteratorum unio” (A mixed substance is the alteration of the mixables (the elements) so that they will be one) [LC 3.9, p. 74].110 His scholastics contemporaries, following Aristotle, commonly agreed that the four elements (earth, air, fi re, and water) are simple bodies, that all other material substances are compound bodies, and that the four elements are the fundamental physical principles in the genera-tion of each compound body. This was also Wyclif ’s view. Further, he explains the process of generation. In the dissolution of compound substances, there is a fi ne division of elements into small parts, and these elemental particles interact, and are mixed together in a regular quantitative proportion, which gives rise to a compound substantial form (also a common scholastic view). Further, Wyclif agrees with his peers that this in fact produces a single compound substance. But scholastics disagreed about how, as Aristotle claims, the elements remain potentially in a mixed body. The evangelical doctor presumes that the traditional three analyses of the moderns, Averroes, and Avicenna are the choices available to reason, and he argues that only one of these, which supports an atomistic account, is a viable alternative.

Wyclif tells us that most of his contemporaries take the modern view that generation of a compound is caused by the interaction of the elements, but, in fact, no elements actually remain when a superadded compound form arises [LC 3.9, p. 76]. For example, the followers of Thomas Aquinas claim that each compound substance is a composite of prime matter and just one substantial form, so, in a new mixed

109 For a detailed discussion of the scholastic problem of how the elements remain in the mixed, see Maier, An der Grenze von Scholastik und Naturwissenschaft, pp. 1–140.

110 Aristotle, On Generation and Corruption, 1, ch. 10, 327b–328b. See also Wyclif, Tr., pp. 88–89: “Mixta autem concipio sic componi, cum secundum Aristotelem mixtio sit mixtibilium alteratorum unio; elementa enim secundum partes indivisibiles vel saltem nobis insensibiles ad invicem commiscentur, et formae superadditae resultant, habita proportione mixtionis suffi cienti ad illam formam superadditam principiandum.”

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body, the elements cannot actually remain with their proper forms. The ontological structure presupposed is inconsistent with the continu-ity of the elements. So, when a compound form is generated, the four elemental forms are concomitantly destroyed.111 Scotus and Ockham agree that inorganic compounds are homogeneous substances, so gold or lead must have just one substantial form throughout. When a com-pound form arises, the elements are destroyed. This became the most prevalent fourteenth-century view.

But Wyclif fi nds this view unacceptable. He argues, fi rst, that this “modern” view is inconsistent with the defi nition of an element, “for an element is the intrinsic cause of a thing.”112 Not unreasonably, Wyc-lif demands that elements actually compose the body that they cause. Second, the ‘modern’ view is inconsistent with experience. “For both in art (i.e., alchemy or chemistry) and in nature it is known that from parts of a compound different elements result, as in the dissolution of stones, the calcination of metals . . .”.113 Wyclif here asserts the following principle of composition: a body must be composed of that into which it is resolved. So compounds must be composed of these elements. This principle reappeared in the seventeenth century in writings, for example, of Daniel Sennert and David Gorlaeus in support of their atomistic views.114 Here Wyclif argues that the corruption of mixed bodies provides evidence that is inconsistent with the “modern” view of how compounds are formed. He contends: if compounds are resolved into the elements, as we fi nd they are in the natural dissolution of putrefying fl esh or fi sh or in chemical or metallurgical processes, then compounds must be actually composed of the elements [LC 3.9, p. 77].115

111 From the Thomistic viewpoint, since a substance can have no more than one substantial form, when the mixed form is generated, the elemental forms are destroyed. The result is that the elements themselves vanish, but they are said to remain ‘virtually,’ for the elemental qualities are a subset of the qualities of the new mixed form. This view of Aquinas, in fact, developed over time. For discussion of this development, see Zavalloni, Richard de Mediavilla et la controverse sur la pluralité des formes.

112 LC 3.9, p. 76: “Nam elementum, ut huiusmodi, est causa intrinseca rei . . .”113 LC 3.9, p. 77: “Nam tam arte quam natura cognoscitur ex partibus mixtorum

elementa dispariter resultare, ut in dissolucionibus lapidum, calcancionibus metal-lorum . . .”

114 For discussion of this seventeenth century development, see my paper “Sennert’s Sea Change: Atoms and Causes.”

115 Wyclif appears here to demonstrate some familiarity with and interest in the alchemical views of his time.


An alternative account, ascribed to Averroes116 is the view that each element actually remains in a compound substance, but the substantial form of each is reduced in intensity [LC 3.9, p. 75]. In the words of Cambiolus of Bologna, a fourteenth-century Averroist: “These four forms are refracted in a certain median form, which informs one mat-ter.”117 Here, just as when blue and yellow interact, each is reduced in intensity and a new form, that of green, is produced, so too when the contrary elements interact, their forms are reduced in intensity to produce a single new median compound form. The elements therefore remain in a compound body, but they remain in an altered state.

Wyclif maintains that this view is also inconsistent, in this case with the nature of substantial forms. Socrates can become more hot in the summer, less hot in the winter, more tanned by the sun, less corpulent in old age, but he cannot become more or less of a human being at any stage of his life. Humanity, unlike hotness or brightness, does not admit of degrees. In Wyclif ’s view, by parity of reason, no substantial form can admit of more or less; fi re cannot become more or less fi ery nor air more or less airy.118 So the Averroist account also will not do.

Wyclif concludes: “I believe (. . .) that the elements are really in a compound, according to their own proper forms and places, as Avicenna says [ primo causarum ca. 3 and many other places].”119 Some Aristote-lians adopted the Avicennist view that the elements actually remain in a compound. For example, both Richard of Middleton (before Wyclif ) and, after Wyclif, Paul of Venice (c. 1369–1429) maintain that the

116 Wyclif, LC 3.9, p. 75: “Patet ista posicio 3 de celo 67.” See Aristotle, De Caelo, Book 3, folio 67, Aristotelis Opera cum Averrois Commentariis [Venice, 1562–1574], vol. 5, p. 227: “Dicemus quod formae istorum elementorum substantiales sunt diminutae formis substantialibus perfectis et quasi esse est medium inter formas, et accidentia, et immo non fuit impose ut formae eorum substantiales ad miscerent, et proveniret ex collectione earum alia forma, sicut, cum albedo et nigredo admiscentur, fi unt ex eis mixti colores medii.”

117 Cambiolus of Bologna, “Utrum elementa maneant in mixto secundum proprias formas aut solum secundum esse virtuale,” Ms. Vat. Ott. 318, ff. 4v–8r, edited by Kuk-sewicz, Averroïsme Bolonais au XIV e Siècle, p. 151: “. . . iste forme quatuor sunt refracte in quandam formam mediam, que informat unam materiam.”

118 Wyclif, LC 3.9, p. 75 “. . . essencia que est forma substancialis, non suscipit magis et minus pocius de substanciis elementaribus quam de mixtis; ut sicut nichil est reliquo magis homo, sic nec aliquid est reliquo magis ignis.”

119 Wyclif, LC 3.9, p. 79: “Credo 3am sentenciam in hac parte; scilicet, quod elementa sunt realiter in mixto secundum situs and formas proprias, ut dicit Avicenna, primo causarum, ca 3, et alibi multis locis.” So Wyclif says (LC 3.9, p. 80) that the juxtaposi-tion of corpuscles (i.e., elemental atoms) constitutes a true compound (“juxtaposicio corpusculorum, ceteris requisitis, constituunt vere mixtum”).

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elemental forms remain extended uniformly throughout a compound substance. In Paul’s description, the elemental forms remain as inhering but not informing [i.e., not determining] forms.120 But Wyclif also fi nds this way of interpreting Avicenna’s view unacceptable. This homoge-neous distribution of the elements throughout a substance will not do, because substantial forms of the same grade cannot be in the same part of matter at the same time. First, Wyclif contends, a substantial form cannot inhere in a substance without informing it. Second, earth and water cannot occupy the same place at the same time.

Instead, Wyclif recommends a solution that he well knew virtually all of his Aristotelian contemporaries found unacceptable. He maintains that the elements remain as minuscule atomic parts, too small to be perceived, so that all compound bodies are in fact heterogeneous. Like seventeenth century atomists, he here distinguishes appearance and reality. The problem that his contemporaries found with the view that he accepted was that the interaction of elements would not produce a compound substance. The claim here is that if air, earth, fi re, and water actually remain “according to their own proper forms and places”121 (as corpuscles or atoms), this would produce a mere mixture of many substances, a collection or a heap. The result would be not a single compound substance (mixtio) which must be homogeneous, but merely an aggregation of corpuscles.

Wyclif explains this objection: “First, it is seen that it is not properly mixed, but is a juxtaposition of corpuscles; that thus, one having the eyes of Lynceus will see in which way whatever element would be in a place separately; and in this way likewise humans and all genera of bodies are mixed in the world, and there would be no superadded sub-stantial form; since no compound would be truly something one.”122 He

120 Paul of Venice, Expositio super libros De genertione et corruptione [Venice, 1498], f. 63a, explains of the elements: “They do not remain as a composition of form with matter, but rather do they remain as forms inhering in matter . . .” (. . . non manent secundum compositione forme cum materiae; manenteni bene inherentia forme ad materia); and, f. 63b–c: “The form of the mixed itself . . . informs and inheres. However other forms, namely of the elements, inhere and do not inform.” (Ipsa forma mixti . . . informat et inheret; alie autem forme ut elemententorum inherent et non informant.”)

121 Wyclif, LC 3.9, p. 79.122 Wyclif, LC 3.9, pp. 79–80: “Primo, videtur quod proprie non sit mixtio, sed iux-

taposicio corpusculorum; quod sic, habens occulos linceos videret quomodo quodlibet elementum foret seorsum positum; et sic per idem homines et omnia genera corporeum essent commixta in mundo, et nulla foret forma substancialis superaddita; cum nullum mixtum foret vere aliquod unum.”


rejects this conclusion: “To that is denied the fi rst consequent (i.e., no superadded substantial form), because the juxtaposition of corpuscles, with other requisites, constitutes a true compound.”123 This is because when cats or cows or human beings are mixed together with others in the world, this is “not towards an end that results in a substantial form constituting a substantial compound of a different species.”124 When elemental corpuscles are mixed together, it is towards such an end.

Hence:

1. Fido, Mungojerry, and Dumbo cannot combine to compose a new substance that has its own specifi c substantial form.

2. But all agree that the elements (earth, air, fi re, and water) can combine in an appropriate proportional relation to produce a new compound substance with its own substantial form.

3. Therefore, elemental corpuscles can do so.

Dogs and cats and elephants are merely collections or aggregates of entities. The elemental corpuscles are not. Unlike a mere aggregate of entities, elements, in an appropriate proportional relation, can compose a distinct compound substance with a higher grade of substantial form. So, Wyclif contends, elemental corpuscles can act “towards an end that results in a (distinct superadded) substantial form constituting a substantial compound of a different species.”125

Wyclif, therefore, claims that in substantial change, the elemental corpuscles interact and, towards the end of generating a new sub-stance, they are mixed together in a regular quantitative proportion. This process produces what Wyclif calls minima naturalia (analogous to molecules),126 and, in each of these, different proportional relations of the elemental particles are naturally associated with and so give rise to different compound substantial forms. The apparent homogeneity of a golden globe is the sameness of a supervening compound form in each compound particle of that substance. He concludes: “whoever speaks

123 Wyclif, LC 3.9, p. 80: “Ad illud negatur prima consequencia, cum iuxtaposicio corpusculorum, ceteris requisitis, constituunt vere mixtum.”

124 Ibid.: “Et sic conceditur homines commisceri ad invicem cum aliis, et propor-cionaliter de ceteris partibus huius mundi, sed non ad fi nem quod forma substancialis resultet constituens mixtum substanciale disparis specie.”

125 Ibid.126 Ibid.: “. . . quod philosophi secundum gradum minimum vocant minimum natu-

rale.”

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truly of a compound [mixtio] ought to concede either small bodies [corpora parva] or the matter of bodies to be juxtaposed, and to be continued by a superadded form.”127

Wyclif here claims a twofold consequence of his Avicennist view, fi rst, there must be orderly structures of contiguous ( juxtaposed) elemen-tal atoms, and second, a mixed body, i.e., each natural minimum (or compound particle), must be determined by a distinct super-added substantial form. This leads him to a view in which, each compound material substance, organic and inorganic, is composed of a plurality of substantial forms. Gold or salt, blood, bones or fl esh are each a single mixed substance because each is composed of homoeomerous minima naturalia. Wyclif ’s world is therefore composed of orderly structures of particles and organized by collections of like substances. Air is each air atom and any collection of these atoms; salt is a natural minimum composed of a particular proportional organization of elemental atoms and a supervening salt form, and salt is any collection of such compound particles. A human being is a particular composite of body and mind, and humanity is the collection of all such substances.

At Creation, God created material indivisibilia, and, in turn, from combinations of these, produced the size and shape of the elemen-tary corpuscles (of earth, air, fi re and water) that compose Wyclif ’s world. This is a world composed of elementary units, and, in turn, of compound units and orderly collections and combinations of these. It seems that Wyclif thought it necessary to accept, on the basis of reason and observation, that the elements fundamental to the formation of a compound body must remain in that body, and he was willing to accept as logical consequences of this view, enduring elemental corpuscles and a plurality of substantial forms in every compound substance. Further, in accordance with his methodology, this conclusion of reason is con-sistent with his fi nitist interpretation of Scripture and his consequent commitment to an atomistic account of the natural world.

4. Conclusion

John Wyclif has long been cast in the role of an innovator, and, in particular, as the morning star of the Protestant Reformation. What I

127 Ibid.: “. . . cum omnes vere loquentes de mixtione oportet concedere vel corpora parva, vel materias corporum, iuxtaponi, et per formam superadditam continuari.”


have tried to show is that this is a limited picture of the innovations of this now little known fi gure. In the context of his Aristotelian ontology, Wyclif provides an interesting atomistic account of the natural world. Further, here too he anticipates some later views, though his particular theory as a whole remains distinctive. For example, like seventeenth century atomist Daniel Sennert, Wyclif maintains that the fundamental building blocks are indivisible atoms and both of these atomists cite Democritus in support of their atomistic views.128 But both depart signifi cantly from Democritean atomism.

Unlike Democritus, both claim that there are several grades of atoms, all of which were formed by God at Creation. Both assume elemental corpuscles (of earth, air, fi re, and water), and both claim that each elemental corpuscle is a substance that is a composite of prime matter and a substantial form. Both also claim that combinations of elemental units, in turn, constitute compound particles, which are hierarchically composed of a structure of elemental corpuscles and a supervening substantial form. Both develop atomistic accounts in the context of Aristotelian hylomorphism and scholastic pluralism (the view that a compound substance can have a plurality of substantial forms).

Sennert, however, praises Democritus, because this ancient atomist realized that since bodies cannot come to be from nothing nor from points, there must be smallest physical particles.129 Sennert’s primitive indivisible units are extended elemental atoms. Wyclif adds a simpler grade of matter. Our Oxford master, rejecting that bodies cannot be composed ultimately of points, makes that very claim. His indivisible atoms, cast as the fundamental building blocks of extended bodies, are not blocks at all. That is, they are not extended physical units; they have no length, breadth, or depth. According to Wyclif, elemental cor-puscles are themselves composed from some structure (known to God) of unextended indivisible atoms, and these elemental corpuscles are the extended atomic units fundamental to all bodies in the world.

To conclude, I have here argued that Wyclif ’s atomism was moti-vated not simply by the ancient atomism of, for example, Democritus or Plato, authorities he cites [Tr p. 84; LC 3.9, p. 132], nor simply by

128 For Daniel Sennert’s atomistic theory, see Daniel Sennert, Hypomnemata Physica, “De Atomis, & Mistione,” [Lyon, 1650], pp. 156–167. See also Michael, “Sennert’s Sea Change: Atoms and Causes.”

129 Sennert, ibid., p. 158: “Cum enim videret Democritus, corpora naturalia neque ex nihilo, neque ex punctis fi eri, necessario statuit, ea ex minimis corpusculis componi.”

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rational arguments and the explanatory power of atomism nor by the evidence of experience, though he draws upon reason and experience to justify his atomistic view. As discussed above, these factors contributed to his development of a corpuscular matter theory, but these were not his principal motivation. Instead, surprisingly enough, Wyclif ’s cor-puscular theory of matter was prompted, perhaps most acutely, by the requirements of his distinctive methodology, which employs Scripture as the source of all truth. Sennert was a physician and a chemist, and his conversion to an atomistic view was infl uenced, to a large extent, by his commitment to the distinctive scientifi c method proposed in Iacopo Zabarella’s logic.130 It is instructive to note that, in Wyclif ’s case, interesting scientifi c developments were inspired, at least in part, by theological motives. These considerations have led me to conclude that Wyclif ’s natural philosophy, though now largely neglected, is in fact of great interest, not simply because of the theories he proposed, but also (in the light of lessons to be learned from history), as an invaluable resource for understanding the problems, perspectives, and strategies infl uencing the development of atomism, and, indeed, the eventual rise of modern science.

130 Iacopo Zabarella, Opera Logica [Venice, 1586]. There were many editions of this work.

BLASIUS OF PARMA FACING ATOMIST ASSUMPTIONS

Joël Biard

Fourteenth-century arts masters offered better and more thorough accounts of atomism than their predecessors, even when they intended primarily to criticize it. The masters knew Democritus only indirectly through Aristotle’s various critical works, especially De generatione et cor-ruptione. But, as Aristotle linked the Democritean conception of elements with the notion of the indivisible,1 the whole study of continuum and indivisibles that has been developed in the thirteenth and fourteenth centuries, both in physical and mathematical contexts, invites us to confront atomism.2 Not only did Thomas Bradwardine provide an overview of the various solutions to problems associated with the con-tinuum,3 but Walter Chatton, Nicholas Bonetus, and Gerard of Odo (if we assume that Nicholas of Autrecourt had no direct infl uence on those debates) made much more consistent the atomist hypothesis, whether one adheres to it or rejects it.

In his various writings on natural philosophy Blasius of Parma often develops issues which are not strictly Aristotelian. A marginal comment in a manuscript copy of the second draft of his commentary on the Physics, raises the question of Blasius’s acquaintance with the atomist thesis:

This question is dealt with in the fi rst book On the Heavens, you also fi nd it determined by Gerard of Odo in a small powerful book, at the

1 See De Generatione et corruptione, I, 1, 314 a 21–24.2 Hence, Democritus is quoted twice in Nicole Oresme’s Questions on De generatione

and corruptione. Concerning the defi nition of the element, he mentions the Aristotelian defi nition according to which the element is that in which the body is reduced. He distinguishes three possible interpretations: “. . . pro cuius expositione est sciendum quod triplex est dissolutio: quedam in partes integrales, et sic intelligit Democritus corpora resolvi in elementa, id est athoma . . .” (Nicole Oresme, Questiones super De generatione et corruptione, [Caroti], pp. 199–200).

3 The text of the Tractatus de continuo has been edited by John Murdoch, in Geometry and the Continuum in the Fourteenth Century, pp. 139–471. Some extracts have been translated into French in Sabine Rommevaux, “Bradwardine: Le continu,” pp. 89–135.

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beginning of which one can fi nd the Questions on the Physics by Thomas the English.4

This marginal note is probably not from Blasius, but rather from the copyist.5 Nevertheless, it invites us to compare the various authors, a comparison that, unfortunately, I won’t be able to carry out in this short chapter. Still, it suggests that atomism constitutes the horizon for these questions about the continuum. At any rate, it makes it clear that atomism cannot be eliminated from the discussion. It is hard to say much else about this remark insofar as it would require a better understanding of Blasius overall conception of matter. Unfortunately, the lack of a critical edition of the Questions on the Physics makes it extremely diffi cult to get any comprehensive picture of the whole set of his conceptions, inasmuch as it is quite possible that Blasius changed his mind on this topic.6

Blasius’s attitude towards atomism, or rather his use of atomist argu-ments, is related to two main problems. The fi rst one concerns the void, a problem I will not deal with in detail in this chapter. On this point, Blasius defends an original position. He denies the real existence of a void while accepting at the same time a hypothetical void as fecund for studying the nature of motion, both in his Questions on Bradwardine’s Treatise on Proportions as well as in his Questions on the Physics. Concern-ing the fi rst treatise, he utilizes the assumption that a void exists when discussing Bradwardine’s rule in order to determine the motion and speed of a body in the absence of a resisting medium.7 This approach allows him to dissociate the temporal factor from the form accomplished in and by a motion (either a qualitative form, in the case of an altera-tion, or distance, in the case of a local motion). He scrutizines this hypothesis much more extensively in the Question on the Physics. Here,

4 “Tangitur etiam hec question I° Celi; habes etiam ipsam determinatam per Gerardum Odonis in libro parvo viridi, in cuius principio sunt Questiones Physicorum secundum Thomam Anglicum,” in Questiones Physicorum, VI, q. 2, Ms. BAV, Vat. Lat. 2159, f. 164va (in margine), quoted according to Federici Vescovini, Astrologia e scienza, p. 340. For the Questiones de Celo, see bk. I, q. 8, in Ms. Milano, Biblioteca Ambrosiana P 120 sup.

5 Bernardus a Campanea di Verona (See Federici Vescovini, loc. cit.; and Caroti, I codici di Bernardo Campagna. Filosofi a e medicina alla fi ne del sec. XIV ).

6 This is Graziella Federici Vescovini’s claim (op. cit.), who argues that Blasius modifi ed or smoothed out his theses between the fi rst and the second redaction of his Questions on the Physics after the condemnation of 1396.

7 See Blasius of Parma, Questiones circa tractatum proportionum magistri Thome Bradwardini [Biard & Rommevaux], pp. 164–166 and p. 174.

blasius of parma facing atomist assumptions 223

an entire question is devoted to the possibility of motion in the void.8 It seems likely that he takes as a starting point the traditional opin-ion—since Ibn Baja—that motion would be instantaneous in a void since, resistance being equal to zero, the ratio of force/resistance would be infi nite. Denying the possibility of an infi nite speed, he transforms this objection against a void into a useful hypothesis. He admits that a motion in the void would be successive. It can therefore be assigned a certain speed and be used in order to compare motion in the void with motion in a given medium.

However interesting this question concerning a hypothetical void may be, it does not seem explicitly connected with atomism. The situation is different in the case of the continuum. It should be noted, however, that by way of the fi rst article in question 1 of book 6 of the Physics (devoted to the continuum) Blasius introduces a problem, albeit quite briefl y, concerning the motion of two mixed bodies—unequal in quantity but similar with regard to their mixture—in the void. While it might simply be a purely scholastic exercise, without any link to the question,9 it may suggest a strong connection between the problem of the void and that of the continuum. Dealing with the continuum, Blasius develops a moderate and complex position on indivisibles.

1. The Composition of the Continuum

Blasius of Parma examines the composition of the continuum in ques-tions 1 and 2 of book 6 of the second version of his Physics commentary. I will deal with some related passages from the fi rst version of this text later.10 In the fi rst question, Blasius asks whether one may conclude with valid reasons that a continuum is composed of indivisibles. The second question tackles the problem from the opposite perspective, asking whether a continuum is infi nitely divisible.

8 Questiones Physicorum (second redaction), IV, 5, Ms. BAV, Vat. Lat. 2159, ff. 122vb–126rb.

9 In some textbooks, we may fi nd sophisms introduced as exercises, but without any link to the topic; this is the case in the Questions on the Logical Tracts, according to a usual practice in Northern Italian Universities. See A. Maierù, “I commenti bolognese ai Tractatus di Pietro Hispano.”

10 The fi rst version corresponds to the lectures given in Padua between 1382 and 1388; we only have the two fi rst books. The second version, of which we have a com-plete text, corresponds to the lectures given in Pavia in 1397. See Federici Vescovini, Astrologia e scienza, p. 430 and pp. 433–434.

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In the fi rst question, Blasius adopts an attitude that can be found in other texts which makes it particularly diffi cult to fi gure out his posi-tions. In the determinatio of the second article of the question, he seems to embrace the opposed views (the same attitude is found, for example, in the lengthy Question on intension and remission of forms).11 This procedure obviously makes it diffi cult to assess which one corresponds to his own position: does he set forth the various possible answers because of his inability to settle the issue? Is he subtly more inclined towards one of the answers without making it explicit (whatever may be his reasons to do so)? Does he implicitly oppose each point of view? With these interpretive problems in mind, here is how he introduces the second article:

I begin the second article, in which I will answer the question in a nega-tive way, and then I will make you understand that the question can be affi rmatively defended.12

The refusal to consider the continuum as composed of indivisibles is based on a very traditional style of argumentation, which from the outset relies on geometry. Indeed, Blasius demands that one accepts a Euclidean principle according to which any line can be divided into two equal parts. Such a claim, as we well know, is particularly problematic if a line is made up of points, above all if they exist in a fi nite number. The argument unfolds in three quick steps.

The fi rst part (arguments 1 and 2) considers the number of points into which a line can be divided: it can be neither even nor odd. The second part claims that a line can neither be made up of an infi nite nor of a fi nite number of points (conclusions 3 and 4). The assump-tion of a fi nite number is invalidated by the preceding arguments. If the number is infi nite, then the line is either fi nite or infi nite. It cannot be infi nite, so it should be fi nite. Again, it is asked whether its half is composed of a fi nite or an infi nite number of points. Finally, arguments 5 and 6 proceed to a quick generalization concerning the composition of a surface from lines, and a body from surfaces.

11 This question has been edited by Federici Vescovini, “La Questio de intensione et remissione formarum de Biagio Pelacani di Parma,” pp. 432–535.

12 “Transeo ad secundum in quo determinabo questionem pro parte negativa et tandem dabo tibi intelligere questionem posse sustentari pro parte affi rmativa,” Ms. BAV, Vat. Lat. 2159, f. 162va.


All of this is nothing but ordinary reasoning against the composition of a continuum from indivisibles and we may even say that he goes far less deeply into the details than did English scholars at the beginning of the fourteenth century. But it is interesting that he introduces—only as a defensible position for the time being—the assumption that the continuum is composed of indivisibles: “Now I present some conclusions which can be defended without contradiction from the other point of view”.13 Blasius then opts for the infi nitist solution outlined in the fi rst fi ve conclusions.

1. “Even if a line were made up of points, we could not infer that it is made up of an odd or even number of points”.14 The purpose is to set aside the aforesaid contradiction between an odd and even number of points, of which the line would be made up in this case. How to proceed from here? At this moment, Blasius takes up the proportional parts of a continuum, which are neither even nor odd.

2. “Even if a line were made up of points, we could not infer that it is made up of a fi nite number of points”.15 This is a consequence of the former conclusion.

3. “If a line were made up of points, it would be made up of an infi -nite number of points”.16 Indeed, if it is made up of points, it will be a fi nite or an infi nite number of points. Yet it cannot be a fi nite number, according to the preceding conclusion.

4. “That any line is made up of an infi nite number of points does not include any contradiction”.17 The rationale is very weak: the opposite cannot be demonstrated, and no contradiction follows. But Blasius signifi cantly adds that many learned people support this claim: “multi sapientes tenuerunt illud”.18 He will come back to this problem later: “To say that a line is made up of points is not

13 “Nunc pono conclusiones pro alia parte defensibiles a contradictione,” ibid., f. 162vb.

14 “licet linea esset ex punctis composita, non ex hoc posset determinari illa esse ex punctis paribus vel imparibus compositam,” ibid. ff. 162va–163ra.

15 “licet linea esset composita ex punctis, non ex hoc determinari posset illam ex punctis fi nitis esse compositam,” ibid. f. 163ra.

16 “si aliqua linea est composita ex punctis, ipsa composita esset ex punctis infi nitis,” ibid., f. 163ra.

17 “quod omnis linea sit composita ex punctis infi nitis contradictionem non includit,” ibid., f. 163ra.

18 Ibid., f. 163ra.

226 joël biard

inconsistent with the principles of the mathematicians, nor with the writings of the philosophers”.19

5. The fi fth conclusion introduces unequal infi nites. Apparently, the conclusion is related to a particular point: it is contradictory to claim that there are as many points in the circumference as in the diameter of a circle since, in this case, they would be equal (as the diagonal and the side of a square, and so on). Therefore, given that an infi nite multitude of points has been accepted, there exists another one infi nitely larger.20 Another consequence follows: “multitude” has to be understood in a broader sense than “number”. An infi nite multitude of points is not a number:

‘Multitude’ is more general than ‘number’, so that any number is a mul-titude, but not conversely. Indeed, the term ‘number’ adds to ‘multitude’ the idea of numerability or being able to be numbered. But here, the term ‘multitude’ does not connote it.21

These passages make plausible the assumption of a composition from an infi nity of points. That assumption remains, however, within the framework of a mathematical discussion, even if the examples are basic. The only way to get out of the contradiction is to suppose another meaning for the infi nite continuum—without any possibility of being precise about that meaning. The idea of point remains rather vague.

Nevertheless, article 3 will attempt to make the hypothesis more reliable by examining some of the diffi culties that result from it.

First, under these conditions, one asks whether a line can be divided into equal parts. If one posits a fi nite number of points, it will generate contradictions, not if one posits an infi nite number of points.

Second, one asks whether it is possible to remove a point from the whole multitude of points in a line. It is answered negatively.

Third, one asks whether a line is resolvable into points. It is answered positively, but it is resolvable only potentially. Here, Blasius may seem to contradict himself or, at least, to attenuate his fi rst aim. Anyway, he

19 “dicere lineam esse compositam ex punctis non repugnat principiis mathemati-corum, nec sermonibus philosophorum,” ibid., f. 163rb.

20 “Ex quibus sequitur quod data multitudine punctorum infi nita est alia maior in infi nitum,” ibid., f. 163ra.

21 “multitudo est superius ad numerum, sic quod omnis numerus est multitudo, et non econtra. Addit enim hic terminus ‘numerus’ supra multitudinem numerabile vel posse numerari. Sed hic terminus ‘multitudo’ non connotat hoc,” ibid., f. 163rb.


clearly wants to preserve an infi nitely divisible continuum for math-ematical objects.

More interesting is the fourth diffi culty. The preceding assumptions being accepted, Blasius asks whether points are immediately adjacent to one another. Here, two alternative answers are indicated: it may be said that a point is immediately adjacent to another, but since it has no part, it touches it as a whole and not by one of its parts (this relates to the problem of contact, something which Blasius paid great attention to in other texts),22 or it may be said that contact only concerns mag-nitudes, in which case a point cannot be said to be touching another, nor to be distant from another.

The following diffi culties and their solutions deal with the notion of limit: fi rstly the limit of a line, and then, quite lengthily, the limit of a quality that would be exclusively limited by another one. I leave out these developments, which would suppose a detailed examination of the status of a quality and of the theory of the intension of forms.

The second question returns to the main problem, but tackling it from the opposite direction. It is asked whether a continuum is infi nitely divisible. It is here that the copyist mentions Gerard of Odo. But the reciprocity between the two questions is only apparent. The fi rst ques-tion stuck with simple mathematical examples (or, in the end, with some considerations on the limit of a quality). Here, we actually fi nd a long set of similar arguments, at the beginning of the part quod non. Indeed, in the second set of arguments (introduced by “Secundo ad principale”), fi fteen are of a rather mathematical sort, and the last three deal with motion. But it is preceded by another shorter series that immerse us in a different clime made of purely physical examples: a wire fence resist-ing a pulling force; water freezing in a clogged vase (curiously enough such a phenomenon is supposed to create void); a ray refracted when passing from one medium to another with a different density.

The question’s determinatio provides a set of terminological clarifi ca-tions, which are taken from the fi rst draft of the text, which I will have to deal with later. The fi rst details concern the categorematic and syn-categorematic meaning of the infi nite, that is to say, its determination

22 See the question “Utrum duo corpora dura possunt se tangere,” mss. Venice, Bibl. Marc. Lat. cl. VI, 155 (3377) = codex 18, Valentinelli IV, 230, ff. 105ra–112ra; Bologna, Bibl. Univ. 2567; a copy in Oxford Canon. Misc. 177, ff. 155ra–158vb gives a different text; but we fi nd the same text in the Questiones de anima, Ms. BAV, Vat. Chig. O. IV. 41, ff. 195ra–197rb, and Naples, Bibl. Naz., VIII. G. 74, ff. 128r–136r.

228 joël biard

according to the place of the term in a sentence and the various defi -nitions that can be given to it. These topics had become almost com-monplaces from the second quarter of the fourteenth century, after the eighteenth sophisma of William Heystesbury’s Sophismata and after John Buridan’s, Gregory of Rimini’s and Albert of Saxony’s work.23

Blasius next provides some more details about the terms “divisible” and “indivisible”. What is important here is the distinction between two meanings of “indivisible”:

In one sense, something is said to be divisible because it is naturally able to be divided, and conversely, something is said to be indivisible because it is not naturally able to be divided. In another sense, something is said to be divisible because, although it is not naturally able to be divided, one can nevertheless think that it is divisible without any contradiction. Hence, the heavens are actually divisible, as well as anything that has parts located one apart from the other.24

In question 10 of book 1 in the fi rst version of his Physics, Blasius makes the same distinction, with some additional details. A one foot piece of earth is used as an example of a thing able to be divided. It is not impossible to divide it into two or three pieces, etc. (by means of an actualis divisio). The sun, the heavens and the eighth sphere are examples of the second sense. But Blasius also presents this distinction between the two senses by opposing the “real division” and the “intel-lectual division”.25

After a last terminological clarifi cation concerning proportionality, the second question continues and establishes conclusions concerning the ratio of the whole to its parts. These conclusions will be put to the test in the third article.26 Curiously enough, Blasius is so concerned about establishing that a line is infi nitely divisible, mostly by means of a sequence of divisions into proportional parts, that he seems to recon-sider what he has established in the preceding question concerning the

23 On this topic, texts from John Buridan and Gregory of Rimini can be read in the textbook De la théologie aux mathématiques, respectively pp. 253–279 and 197–219. See also, Joël Biard, “Albert de Saxe et les sophismes de l’infi ni,” pp. 288–303.

24 “Uno modo dicitur aliquid divisibile quia illud est aptum natum dividi, et per oppositum aliquid dicitur indivisibile quia non est aptum natum dividi. Alio modo dicitur aliquid divisibile quia licet non sit aptum natum dividi potest tamen sine con-tradictione intelligi quod sit divisibile. Et sic celum bene est divisibile, et omne habens partem situaliter extra partem”, Ms. BAV, Vat. Lat. 2159, f. 166vb.

25 Ms. BAV, Vat. Chigi O IV 41, f. 245ra.26 Some passages examine in particular the summons of proportional parts.


composition from points. Conclusions 1, 2, and 3 assert that a line is not made up of points, neither fi nite nor infi nite in number.27 However, this contradiction should be taken as relative. In the fi rst question, one considers composition and makes the assumption that a line is composed of an infi nity of points. In the second question, one considers division and insists on the infi nite divisibility, from a geometrical point of view as well as from a physical one, as we will see.

Now, about this point, we fi nd once again an argument Blasius had used in question 10 of book 1 in the fi rst draft. Before considering infi nite division into proportional parts, he sets out as a preliminary condition that a line is not composed of points, neither a fi nite nor an infi nite number. In the fi rst draft, Blasius’s position was not limited to the theses supported in this question: question 11 asks whether there are limits to the greatness and smallness of natural bodies.

2. Indivisibles and Minima Naturalia

The fi rst version of Blasius’s Physics, question 10, concerns the division of a continuum. This is immediately followed by a question about the limits that one should assign to the greatness and smallness of natural bodies. Even if the conceptual context is quite different, we should take the proximity of these two questions quite seriously. Before considering this problem, it is necessary to examine another text, one of those where Blasius explicitly deals with the indivisible.28 This text is question 15 of book 1 On Generation and Corruption where it is asked: “Can an indi-visible be altered?”29 The question has undoubtedly a twofold motive. First, having no parts, it seems that an indivisible cannot move, neither locally nor qualitatively. Again, the question of the relation between a quality and a subject is raised, whether the subject be imagined as indivisible (as a point) or as something indivisible informing a material and divisible subject (as the soul). For this reason, the text supplements other passages where the soul is conceived as an indivisible.

27 Ms BAV, Vat. Lat. 2159, f. 167ra.28 Blasius also deals more or less directly with the indivisible in his Commentary on the

Heavens, and his Questions on the Soul.29 “Consequenter queritur utrum indivisibile possit alterari” (Questiones de generatione

et corruptione, I, q. 15, ms. BAV, Vat. Chigi O IV 41, f. 23rb sqq.)

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The terminological clarifi cations given in the fi rst article will be retained here: a point can be conceived in two different ways. On this occasion, Blasius uses the verb “ymaginari”:

First article: I notice that we can imagine points in a continuum in two ways. In the fi rst sense, they have a position in a continuum; in the second they do not. An example of the fi rst acceptation: if a line were made up of points. An example of the second acceptation: the intellective soul, which is in the human body although it does not have a position and a place in the human body.30

This distinction goes beyond the difference between the two meanings of the division presented in the second redaction of the Physics, namely the real and the intellectual one. There, Blasius argued that the Heavens could not be divided, although one could distinguish in them some dif-ferences in position. Here, it is about a reality that is not susceptible of division, neither real nor imaginary, since by itself, without considering the body it animates, it has no parts, in whatever sense. Blasius makes it clear that this meaning will not be relevant in the remainder of the question: “. . . this distinction is made insofar as the question only deals with an indivisible which has a position in a continuum”.31 Therefore, in this question, he does not really consider whether a spiritual indivisible, such as the intellective soul, can be altered—even if he incidentally men-tions the classic problem of whether a soul could suffer in the infernal fi re.32 Following G. Federici Vesconvini’s analysis, it seems that in the Questions on the Heavens, Blasius considers the soul as infi nitely divisible from a physical point of view. The only relevant point for our present matter is once again Blasius’s claim that everything is divisible from a physical point of view:

30 “Pro primo articulo noto <quod> duplicter possumus ymaginari puncta in con-tinuo. Uno modo quod habeant positionem in continuo, secundo modo quod non. Exemplum de primo si linea componeretur ex punctis. Exemplum de secundo sicut de anima intellectiva que est in corpore humano licet non habeat positionem et situm in corpore humano,” Questiones de generatione et corruptione, I, q. 15, Ms. BAV, Vat. Chigi O IV 41, f. 23rb.

31 “ista distinctio posita est pro tanto quia questio querit solum de indivisibile habente positionem in continuo,” ibid., f. 23rb.

32 Ibid., f. 24rb–va.


Given any thing that is one, it is divisible since it is a natural thing, and the intellective soul is no exception since, from a physical point of view, it is not indivisible.33

On the other hand, the mathematical concept of the indivisible may be applied mutatis mutandis to the soul which is compared to a point in the De generatione et corruptione. Like a prime number, the intellective soul would be only divisible by itself or by the unit. For this reason, it would be considered as a ratio of equality (ratio equalitatis).

The above-mentioned question from De generatione et corruptione tends to show the contradiction that might result from the possibility of a physical indivisible:

Secondly I notice that, for the question asked, in no way one should imagine that indivisibles have a position in a continuum. However, we want to seek and examine some diffi culties that would follow from such an thought experiment (tali ymaginatione).34

Does it mean that Blasius denies any form of physical minima? It is well known that since the thirteenth century minima naturalia have sometimes been distinguished from indivisibles. This notion fi nds its origin in a passage from Aristotle’s Physics where he objected to Anaxagoras’s idea that the constituent parts of a natural whole cannot be of any size what-soever.35 Averroes developed the idea and Roger Bacon and Albert the Great introduced it to the Latin world. It continues to appear up to the Renaissance, especially in texts of Italian natural philosophy. The natural minimum is determined by the nature of each substance. However, the status of this minimum can vary: sometimes it is the minimum capable of producing an effect, sometimes it is the minimum perceptible by the senses. As a consequence, the concept of a natural miminum opened itself to various kinds of philosophical issues. Duns Scotus denied the opposition between quantum and naturalia, and he criticized those who

33 “quacumque re data que est una, illa est divisibilis quia est una naturalis res, nec fi at instantia de anima intellectiva quia, physice loquendo, ipsa non est indivisibilis,” See Questiones de Celo, I, qu. 9, Ms. Milano, Ambrosiana P 120 sup, f. 23ra—according to Federici Vescovini, op. cit., p. 338, n. 32.

34 “Noto secundo quod nullo modo propter questionem propositam ymaginandum est quod aliqua indivisibilia habeant positionem in continuo. Tamen volumus querere et videre aliquas diffi cultates sequentes ex tali ymaginatione,” Questiones de generatione et corruptione, I, 15, Ms. BAV, Vat. Chigi O IV 41, f. 23va.

35 Aristotle, Physics, I, 4, 187 b 13–188 a 5. On the medieval history of this notion, see John E. Murdoch, “The Medieval and Renaissance Tradition of minima natura-lia,” pp. 91–131.

232 joël biard

accepted some physical minima, while admitting at the same time their incompatibility with mathematics.36 As for Blasius, he will not set the mathematical point of view against the physical point of view. As we have already seen, matter is infi nitely divisible. On the other hand, he introduces the idea of a necessary proportions among natural beings, along with the idea of limits beyond and below which a natural being ceases to exist. This is the subject matter of question 11 of book 1 of Blasius’s Physics in its fi rst redaction: “Eleven, we ask whether any natural body is limited by its greatness and shortness”.37

In this question, Blasius does not talk about indivisibles but only about natural limits. We are leaving the fi eld of mathematical indivisibilism, but on the other hand, if the term ‘atom’ is not used in this context, we come very close to the idea of the smallest part of a body that can exist, including the elements. The problem is raised from the beginning of the fi rst article:

As regards the fi rst article, here is a fi rst remark: that a body be limited in greatness and smallness is equivalent to the fact that in relation to it there should be two limits, of greatness and smallness, which cannot be naturally exceeded.38

Beyond the results expressed in such and such a conclusion, it is the general thought process that is interesting. The limit can be considered from the point of view of greatness and smallness or from the point of view of active and passive powers. Greatness and smallness clearly imply the consideration of matter or subject, and on the other hand, the powers imply the consideration of the relationship between matter and form, and this from two different points of view: either from the point of view of the introduction of a form into matter, or from the point of view of the conservation of a substantial form. In the latter case, Blasius still further differentiates the case of a substantial form educed or likely to be educed from matter, and the case of the intel-lective soul coming to animate matter.

36 See Ordinatio [Balic], II, dist. 2, p. 2, qu. 5, pp. 305–306.37 “Queritur undecimo utrum omne corpus naturale sit in magnitudine et parvi-

tate limitatum,” Quaestiones Physicorum (1a redactio), ms. BAV, Vat. Chigi, O IV 41, f. 250ra.

38 “Quantum ad primum articulum sit primum notabile quod corpus esse in magnitudine[m] <et> parvitate limitatum est idem quod respectu eius sint duo termini magnitudinis et parvitatis quod egredi naturaliter non contingat,” ibid., f. 250va.


Here, I only reconstruct the main theses regarding the minimum. The general tendency is to deny a minimum to simple elements, considered in themselves. This is the case in the fi fth and sixth conclusions dealing with the example of fi re. Blasius will later contend that this holds for any element:39 “there is no minimum of fi re since for any given fi re, its half will be smaller”.40 Here we do not consider the existing fi re, but a conceivable or imaginable one. On the one hand, it seems infi nitely divisible. On the other hand, there exists a fi re smaller than any other: “There is a minimum of fi re existing by itself and separately”.41

More interesting is the position of a minimum fi re between certain limits in a given medium. Here we catch a glimpse of the motives behind Blasius’s thought: he wants to introduce ratios between a minimum of matter, a medium, and, possibly, relations of power and resistance. “There is a minimum of fi re that can subsist in such a way and such a time in this medium”.42 The conclusions are reminiscent of the various cases according to which such an element can subsist or be corrupted, in such and such a medium and such and such a time.

However, we still remain within the framework of infi nite divisions. The case is different for mixed bodies which, of course, constitute the most important part of our world. In this case, the connections to be kept gain more and more importance. We will fi rst consider unanimated mixed bodies. They can be treated as simple bodies from the standpoint of permanency and decay: “In no time a mixed and unanimated whole can subsist in a medium”.43 The aim is actually to prevent permanency. But a difference should be noted between unanimated mixed bodies and animated mixed bodies. Paradoxically, the fi rst ones are modifi ed or destroyed if we remove a part. For example, if we remove a part of a stone, we will have another stone or no stone at all. The second ones can subsist if we remove a part, for example, if we remove Socrates’s hand. They are destroyed when the whole is destroyed. Hence, in the case of animated bodies or animals, there is a given maximum and a

39 “et notetis quod in huiusmodi conclusionibus nominamus ignem sed quicquid est dictum de ignis, intelligatis de quocumque alio simplici corpore,” ibid. f. 251rb.

40 “Non datur minimus ignis quia quocumque dato eius medietas est minor,” ibid.41 “Datur minimus ignis per se seorsum existens,” ibid.42 “Datur minimum ignis qui in hoc medio potest sic et taliter permanere per

tantum tempus,” ibid.43 “Per nullum tempus potest aliquod mixtum totum inanimatum in aliquo medio

permanere,” ibid.

234 joël biard

given minimum. But there is no absolute minimum if we consider that a part of the animal is animated.

3. Conclusion

The sequence of questions in the fi rst redaction of the Questions on the Physics is not meaningless. After studying the mathematical division of the continuum, Blasius of Parma wonders whether there are physical minima. It is diffi cult to class him within a predefi ned grid of positions about indivisibles. Blasius is certainly not an atomist in the strict sense, since matter is infi nitely divisible according to him. Nevertheless, like some of his contemporaries, he cannot avoid problems connected with indivisibility. He addresses these problems in various places, defending some positions that at fi rst glance may even seem contradictory. In fact, his goal is to present various points of view. But unlike his habits on other occasions, the aim is not only to contrast the mathematician’s and the physicist’s differing points of view. His distinctions cross these two disciplinary fi elds.

From a mathematical point of view, the composition and the division of a continuum are to be treated differently. Concerning composition, two different claims have been developed. The fi rst one takes up the traditional geometrical criticism of indivisibles. The other one is more original and admits composition from an infi nite number of indivisibles. This non-contradictory assumption is acceptable in mathematics pro-vided that one redefi nes some concepts, such as the concept of whole and parts. Concerning division, the infi nite division of mathematical objects must be accepted.

From a natural point of view, the infi nite divisibility of matter must be admitted without reserve. But natural bodies are composed of matter and form. Some connections are established and must be maintained for the generation and permanency of such and such natural bodies. These connections are not only the more relevant and interesting ele-ments to be studied, they also defi ne the limits within which any such being is conceivable. It is all the more true if the form in question is a soul, since it is indivisible when it is considered in its simple con-nection to itself, and divisible when it is considered as the form of a natural body.

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INDEX OF ANCIENT, MEDIEVAL AND RENAISSANCE AUTHORS

Adam Buckfi eld 58, 236, 245Adam of Marsh 153Adam of Wodeham 16, 40, 88, 127,

130, 139, 171, 182, 236, 238, 240, 244

Albert of Saxony 18, 23, 25, 228, 235, 236

Albert the Great 5, 42, 112, 114, 116, 235, 236

Alexander of Aphrodisias 22Al-Ghazali (Algazel) 7, 19, 66, 68, 87,

236, 245Anaxagoras 15, 112, 113, 114, 231Aristotle 2, 3, 5, 6, 7, 8, 9, 10, 11, 12,

15, 17, 18, 19, 21, 22, 24, 25, 27, 30, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 44, 45, 47, 48, 50, 51, 52, 53, 57, 58, 63, 64, 73, 77, 83, 85, 88, 89, 107, 108, 109, 110, 111, 112, 113, 114, 116, 117, 123, 124, 126, 127, 129, 130, 134, 136, 137, 140, 141, 142, 143, 144, 147, 153, 160, 161, 162, 170, 184, 188, 190, 192, 199, 203, 211, 213, 215, 221, 231, 236

Augustine 4, 14, 53, 55, 202, 203, 204, 206

Averroes 26, 111, 191, 196, 213, 215, 231

Avicenna 10, 19, 87, 213, 215, 216, 236

Bede the Venerable 3Benedict Hesse 84Blasius of Parma 7, 14, 221–234Bonaventure 22, 23, 236

Cambiolus of Bologna 215Campanus of Novara 35, 36, 74Cicero 3, 6, 114, 236

Daniel Sennert 186, 188, 189, 214, 219, 220, 236

David of Dinant 5Democritus 1, 2, 3, 5, 6, 7, 12, 17, 18,

27, 69, 90, 105, 108, 112, 113, 114, 116, 117, 123, 136, 139, 146, 153, 161, 162, 208, 209, 212, 219, 221

Diogene Laertius 34

Epicurus 3, 4, 5, 12, 17, 34, 90, 146, 208,

Etienne Gaudet 162, 165, 166, 208Euclid 10, 19, 20, 21, 22, 26, 35, 36,

67, 70, 72, 83, 237

Filipo Fabri 195Franscesco Patrizi 148

Gaetano of Thiene 138Galileo Galilei 15, 23, 39, 237, 243Gerard of Cremona 162Gerard of Odo 6, 7, 10, 11, 16, 17,

18, 25, 26, 31, 34, 85–106, 107, 127, 138, 140, 141, 142, 160, 170, 171, 174, 183, 211, 221, 222, 227, 235, 240, 245, 246

Giuseppe Veronese 25, 237Godfrey of Fontaines 36, 37, 101, 237Gregory of Rimini 23, 31, 37, 140,

160, 228, 237

Henry of Harclay 7, 9, 10, 14, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 30, 31, 35, 40, 96, 127, 134, 136, 139, 140, 141, 142, 143, 161, 170, 171, 235, 244

Iacopo Zabarella 220, 237Ibn Baja 223Isaac Israeli 162, 237Isidore of Seville 3, 15

Jan Hus 186John Buridan 12, 13, 15, 20, 24, 25,

33, 40, 84, 85, 125, 126, 158, 163–182, 228, 235, 237, 240, 245, 246

John Dorp 40John Duns Scotus 7, 10, 17, 19, 46,

66, 67, 68, 69, 70, 82, 87, 159, 162, 190, 195, 196, 231, 237, 240, 245

John Gedeonis 16, 86John Major 84, 237John Quidort 193

John of Jandun 189John of Mirecourt 37, 244John of Ripa 37, 235, 237John of Salisbury 4John Pecham 189, 192John Tarteys 18, 235John the Canon 16, 31, 87, 138, 235,

237John Wyclif 6, 7, 13, 16, 34, 153,

154, 161, 183–220, 237, 242, 243, 245

Julius Caesar Scaliger 186, 237

Lactantius 4, 237Leucippus 3, 69Lucretius 3, 4, 5, 6, 12, 17, 22, 237

Marbode of Rennes 4, 237Marc Trivisano 86Michel of Montecalerio 7, 12, 13,

158, 163–182, 235, 240Moses Maïmonides 12, 108, 116, 117,

119, 123, 152, 153, 162, 235, 237, 242

Nicholas of Autrecourt 1, 6, 7, 11, 14, 16, 34, 86, 105, 106, 107–126, 127, 138, 149, 153, 157, 221, 238, 240, 241, 244

Nicholas Bonetus 16, 18, 86, 87, 105, 107, 127, 138, 238

Nicole Oresme 18, 23, 25, 37, 64, 221, 235, 238, 240, 243

Parmenide 15Paul of Venice 215, 216, 238Peter Abelard 8, 238Peter Ceffons 37, 243, 244Peter Lombard 41, 81, 87, 127Peter of John Olivi 64, 238Peter of Spain 32, 238Philoponus 22, 238Plato 2, 3, 7, 10, 14, 18, 27, 76, 209,

219Plutarch 22Poggio Bracciolini 3Prisician 3Proclus 22

Rhaban Maur 3Richard Fishacre 49, 238

Richard Fitzralph 66, 246Richard Kilvington 7, 32, 65–84, 235,

241, 242Richard Knapwell 193Richard of Middleton 189, 192, 214,

215, 246Richard Rufus 3, 7, 9, 39–64, 235,

238, 242Robert Grosseteste 5, 23, 40, 47, 50,

53, 141, 238Robert Kilwardby 32, 189Roger Bacon 5, 69, 231, 238Roger Roseth 16, 83, 84, 238Roger Swinshead 37

Theophrastus 18Thomas Aquinas 5, 13, 152, 189, 190,

191, 192, 194, 196, 213, 214, 238, 239, 240

Thomas Bradwardine 7, 16, 18, 19, 20, 23, 24, 26, 27, 32, 33, 34, 36, 37, 66, 75, 82, 87, 221, 222, 235, 236, 238, 240, 241, 243, 244, 245

Thomas of York 153

Urso of Salerno 5

Vincent of Beauvais 4

Walter Burley 7, 16, 22, 31, 66, 100, 145, 163, 172, 173, 175, 178, 180, 181, 238, 239

Walter Chatton 7, 10, 16, 17, 19, 20, 86, 87, 88, 105, 127, 136, 139, 140, 159, 182, 183, 211, 221, 238, 239, 246

William Crathorn 6, 7, 12, 33, 34, 127–162, 211, 235, 238, 241

William Heytesbury 32, 66, 228, 246William de la Mare 192, 193, 238William of Alwick 16, 27, 28, 96, 235William of Champeaux 8William of Conches 2, 3, 5, 15, 238William of Ockham 16, 23, 24, 30,

40, 56, 66, 67, 68, 70, 71, 75, 82, 116, 127, 145, 190, 196, 197, 210, 238, 240, 241, 243, 244

William of Mackesfi eld 193

Zeno (of Elea) 10, 30, 83

248 index of ancient, medieval and renaissance authors

Ebbs, G. 61Elders, L. 239, 240

Federici-Vescovini, G. 222, 223, 224, 230, 231, 240

Frede, M. 42, 236Freudenthal, G. 152, 241Fruteau de Laclos, F. 107, 241

Gauthier, R.-A. 236Gimaret, D. 151, 152, 241Glorieux, P. 192, 193, 236, 238Gnassounou, B. 120, 241Goddu, A. 75, 241Gorlaeus, D. 214Grant, E. 19, 96, 123, 126, 147, 148,

241Grellard, C. 6, 11, 106, 107, 110, 120,

123, 126, 138, 153, 157, 241, 242

Hallamaa, O. 84

Ibrahim, T. 152

Jacquart, D. 5, 121Jolivet, J. 151Jung-Palczewska, E. 66

Kaluza, Z. 107, 110, 113, 115, 116, 118, 123, 156, 163, 164

Kennedy, L. 108Kenny, A. 183, 185, 186, 206Kistler, M. 120Kluxen, W. 152, 153Knuuttila, S. 32Kretzmann, B. 18, 66Kretzmann, N. 10, 18, 66, 154, 183,

208, 210Kuksewicz, Z. 215

Lamy, A. 145, 175, 181Lasswitz, K. 1Lawlor, K. 61Lear, J. 40, 41, 42, 45Leff, G. 186Leijenhorst, C. 243Lévy, T. 245

Allard, G.H. 239, 244Alliney, G. 239, 242Anawati, G.C. 5, 152, 239Annas, J.E. 39, 239Argerami, O. 23, 240Ashworth, E.J. 40, 239Asztalos, M. 239

Bachelard, G. 2, 239Baffi oni, C. 239Bakker, P.J.J.M. 85, 239Barnes, J. 41, 236Bazan, C. 236Bentley, R. 23Biard, J. 14, 134, 145, 164, 222, 228,

236, 239, 241, 245Boas, G. 86, 239Boehner, Ph. 29, 239Bostock, D. 42, 236Braakhuis, H.A.G. 245Brancacci, A. 6, 239Burns, S. 61

Caroti, S. 66, 221, 222, 231, 239, 240, 241, 243

Celeyrette, J. 7, 12, 13, 134, 158, 164, 239, 240, 241, 245

Chandler, B. 240Cohen, I.B. 23, 240Conti, A.D. 183, 240Cosman, M. 240Courtenay, W.J. 66, 127, 163, 164,

166, 240Cote, A. 134, 240Cova, L. 239, 242Crombie, A.C. 240Cross, R. 196, 240

Dales, R.C. 23, 240De Rijk, L.M. 8, 32, 85, 238, 240Dhanani, A. 151, 240Dijksterhuis, E.J. 188, 189, 240Donati, S. 41, 240Duhem, P. 15, 86, 87, 240Dumont, S. 101, 240Dutton, B. 116, 240Dziewicki, M.H. 185, 237

INDEX OF MODERN AND CONTEMPORARY AUTHORS

Lohr, C.H. 85Lüthy, C. 1

Mabilleau, L. 1Machamer, P.K. 243Maddy, P. 60Maier, A. 15, 65, 66, 86, 87, 213Maierù, A. 223Marmo, C. 243Mazet, E. 37McCord Adams, M. 190, 196, 210McVaugh, M.R. 245Menard, J. 239Mendell, H. 40Mendelsohn, E. 243Meyerson, E. 107Michael, B. 163, 164Michael, E. 6, 13, 14, 149, 154, 155,

156, 161, 183, 219Michalski, K. 163Molière 120Molland, G. 5, 36Morel, P.-M. 6Murdoch, J.E. 1, 2, 3, 5, 6, 8, 15, 16,

18, 20, 23, 26, 32, 35, 37, 40, 64, 66, 74, 75, 86, 87, 88, 96, 98, 105, 107, 123, 127, 134, 137, 139, 141, 143, 149, 153, 162, 170, 182, 221, 231

Newman, W.R. 1, 23Newton, I. 1, 113Nuchelmans, G. 164

O’Connor, T. 61O’Donnell, J.R. 106, 149

Pabst, B. 1, 15, 154, 183Panaccio, C. 127Pasnau, R. 127Patzig, G. 42Perler, D. 127Philippe, J. 3, 4Pines, S. 151Pini, G. 61

Podkoński, R. 10, 66, 68, 75Powell, H. 58Pyle, A. 1, 113, 153

Rashed, R. 245Read, S. 245Richter, V. 24, 129Robert, A. 6, 12, 127, 211Roberts, L.D. 245Roensch, J. 192Rommevaux, S. 221, 222Rosier-Catach, I. 145

Sagadeyef, A. 152Sergeant, S.D. 183, 245Schabel, C. 65, 85Schepers, H. 127Sergeant, L. 183Smith, B. 128Snyder, J. 61Sorabji, R. 60Sylla, E. 245Synan, E. 88, 96, 139, 149, 182

Thijssen, J.M.M.H. 84, 85, 174Thomson, S.H. 208Thomson, W.R. 154, 183

Van Melsen, A.G. 1

Walsh, J.J. 85Walsh, K. 66Weijers, O. 85, 166White, M. 64Whitehead, A.N. 25Wilson, C. 32Wolfson, H.A. 151Wood, R. 3, 9, 30, 130, 141, 171Workman, H.B. 183, 185

Zavalloni, R. 189, 192, 214Zubov, V.P. 12, 86, 87, 105, 163, 165,

166

250 index of modern and contemporary authors

History of Science

and Medicine Library

Medieval and

Early Modern Science

Subseries Editors: J.M.M.H. Thijssen and C.H. Lüthy

1. FRUTON, J.S. Fermentation: Vital or Chemical Process? 2006. ISBN 978 90 04 15268 72. PIETIKAINEN, P. Neurosis and Modernity. The Age of Nervousness in

Sweden. 2007. ISBN 978 90 04 16075 03. ROOS, A.M. The Salt of the Earth. Natural Philosophy, Medicine, and

Chymistry in England, 1650-1750. 2007. ISBN 978 90 04 16176 44. EASTWOOD, B.S. Ordering the Heavens. Roman Astronomy and Cosmology

in the Carolingian Renaissance. 2007. ISBN 978 90 04 16186 3 (Published as Vol. 8 in the subseries Medieval and Early Modern Science)

5. LEU, U.B., R. KELLER & S. WEIDMANN. Conrad Gessner’s Private Library. 2008. ISBN 978 90 04 16723 0

6. HOGENHUIS, L.A.H. Cognition and Recognition: On the Origin of Movement.Rademaker (1887-1957): A Biography. 2009. ISBN 978 90 04 16836 7

7. DAVIDS, C.A. The Rise and Decline of Dutch Technological Leadership. Tech nol-ogy, Economy and Culture in the Netherlands, 1350-1800 (2 vols.). 2008. ISBN 978 90 04 16865 7 (Published as Vol. 1 in the subseries Knowledge Infrastructure and Knowledge Economy)

8. GRELLARD, C. & A. ROBERT (eds.). Atomism in Late Medieval Philosophy and Theology. 2009. ISBN 978 90 04 17217 3 (Published as Vol. 9 in the subseries Medieval and Early Modern Science)

9. FURDELL, E.L. Fatal Thirst. Diabetes in Britain until Insulin. 2009. ISBN 978 90 04 17250 0

Published previously in the Medieval and Early Modern Science book series:

1. LÜTHY, C., J.E. MURDOCH & W.R. NEWMAN (eds.). Late Medieval and Early Modern Corpuscular Matter Theories. 2001. ISBN 978 90 04 11516 3

2. THIJSSEN, J.M.M.H. & J. ZUPKO (eds.). Metaphysics and Natural Philosophy of John Buridan. 2001. ISBN 978 90 04 11514 9

3. LEIJENHORST, C. The Mechanization of Aristotelianism. The Late Aristotelian Setting of Thomas Hobbes’ Natural Philosophy. 2002.

ISBN 978 90 04 11729 7

4. VANDEN BROECKE, S. The Limits of Influence. Pico, Louvain, and the Crisis of Renaissance Astrology. 2002. ISBN 978 90 04 13169 9

5. LEIJENHORST, C., C. LÜTHY & J.M.M.H. THIJSSEN (eds.). The Dynamics of Aristotelian Natural Philosophy from Antiquity to the Seventeenth Century. 2002. ISBN 978 90 04 12240 6

6. FORRESTER, J.M. & J. HENRY (eds.). Jean Fernel’s On the Hidden Causes of Things. Forms, Souls, and Occult Diseases in Renaissance Medicine. 2005. ISBN 978 90 04 14128 5

7. BURTON, D. Nicole Oresme’s De visione stellarum (On Seeing the Stars). A Critical Edition of Oresme’s Treatise on Optics and Atmospheric Refraction, with an Introduction, Commentary, and English Translation. 2006. ISBN 978 90 04 15370 7

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Atomism in Late Medieval Philosophy and Theology